RD Sharma solutions for Mathematics [English] Class 10 chapter 5 - Arithmetic Progression [Latest edition]

Write the first three terms in each of the sequences defined by the following

a_n = 3n + 2

Write the first five terms of the following sequences whose nth term are:

`a_n = (n - 2)/3`

Write the first five terms of the following sequences whose nth terms are:

`a_n = 3^n`

Write the first five terms of the following sequences whose nth terms are:

`a_n = (3n - 2)/5`

Write the first five terms of the following sequences whose nth terms are:

an = (−1)ⁿ, 2ⁿ

Write the first five terms of the following sequences whose nth terms are:

`a_n = (n(n - 2))/2`

Write the first five terms of the following sequences whose nth terms are:

a_n = n² − n + 1

Write the first five terms of the following sequence whose nth term is:

an = 2n² − 3n + 1 

Write the first five terms of the following sequences whose nth terms are:

`a_n = (2n - 3)/6`

Find the indicated terms in the following sequences whose nth terms are:

a_n = 5n - 4; a₁₂ and a₁₅

Find the indicated terms in each of the following sequences whose nth terms are:

`a_n = (3n - 2)/(4n + 5)`; `a_7 and a_8`

Find the indicated terms in each of the following sequences whose nth terms are

a_n = n (n −1) (n − 2); a₅ and a₈

Find the indicated terms in the following sequences whose nth terms are:

a_n = (n − 1) (2 − n) (3 + n); a₁, a₂, a₃

Find the indicated terms of the following sequences whose nth terms are:

`a_n = (-1)^2 n; a_3, a_5, a_8`

Find the next five terms of the following sequences given by:

`a_1 = a_2 = 2, a_n = a_(n - 1) - 3, n > 2`

Find the next five terms of the following sequences given by

a₁ = a₂ = 2, a_n = a_n−1 − 3, n > 2

Find the next five terms of the following sequences given by:

`a_1 = -1, a_n = (a_n - 1)/n, n>= 2`

Find the next five terms of the following sequences given by:

a₁ = 4, a_n = 4a_n−1 + 3, n > 1.

Show that the sequence defined by a_n = 5n −7 is an A.P, find its common difference.

Show that the sequence defined by a_n = 3n² − 5 is not an A.P

The general term of a sequence is given by a_n = −4n + 15. Is the sequence an A.P.? If so, find its 15th term and the common difference.

Write the sequence with nth term:

a_n = 3 + 4n

Write the sequence with nth term a_n = 5 + 2n

Write the sequence with nth term a_n = 6 − n

Write the sequence with nth term: a_n = 9 − 5n

is the Consider the expression a_n = 2n − 1, AP . 

is the Consider the expression a_n = 3n² + 1,AP .

is the Consider the expression a_n = 1 + n + n², AP .a

For the following arithmetic progressions write the first term a and the common difference d:

 −5, −1, 3, 7, ...

For the following arithmetic progressions write the first term a and the common difference d:

`1/5, 3/5, 5/5, 7/5`

For the following arithmetic progressions write the first term a and the common difference d:

0.3, 0.55, 0.80, 1.05, ...

For the following arithmetic progressions write the first term a and the common difference d:

−1.1, −3.1, −5.1, −7.1, ...

Write the arithmetic progressions write the first term a and common difference d is as follows:

a = 4,d = -3

Write the arithmetic progressions write the first term a and common difference d is as follows:

`a = -1, d = 1/2`

Write the arithmetic progressions write the first term a and common difference d is as follows:

a = -1.5, d = -0.5

In the following situations, the sequence of numbers formed will form an A.P.?

The cost of digging a well for the first metre is Rs 150 and rises by Rs 20 for each succeeding metre.

In the following situation, involved make an arithmetic progression? and why?

The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.

Find the common difference and write the next four terms of each of the following arithmetic progressions:

1, −2, −5, −8, ...

Find the common difference and write the next four terms of each of the following arithmetic progressions:

0, −3, −6, −9, ...

Find the common difference and write the next four terms of each of the following arithmetic progressions:

`-1, 1/4, 3/2 .....`

Find the common difference and write the next four terms of the following arithmetic progressions:

`-1, (-5)/6, (-2)/3`

Prove that no matter what the real numbers a and b are, the sequence with the nth term a + nb is always an A.P. What is the common difference?

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

3, 6, 12, 24, ...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

0, −4, −8, −12, ...

Which of the following sequences are arithmetic progressions . For those which are arithmetic progressions, find out the common difference.

`1/2, 1/4, 1/6, 1/8 ......`

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

12, 2, −8, −18, ...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

3, 3, 3, 3, .....

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

p, p + 90, p + 180 p + 270, ... where p = (999)⁹⁹⁹

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

1.0, 1.7, 2.4, 3.1, ...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

−225, −425, −625, −825, ...

Which of the following sequences are arithmetic progressions? For those which are arithmetic progressions, find out the common difference.

10, 10 + 2⁵, 10 + 2⁶, 10 + 2⁷,...

Which of the following sequences is arithmetic progressions. For is arithmetic progression, find out the common difference.

a + b, (a + 1) + b, (a + 1) + (b + 1), (a + 2) + (b + 1), (a + 2) + (b + 2), ...

Which of the following sequences is arithmetic progressions. For is arithmetic progression, find out the common difference.

1², 3², 5², 7², ...

Which of the following sequences is arithmetic progressions. For is arithmetic progression, find out the common difference.

1², 5², 7², 73, ...

Find the common difference of the A.P. and write the next two terms  51, 59, 67, 75, ..

Find the common difference of the A.P. and write the next two terms 75, 67, 59, 51, ...

Find the common difference of the A.P. and write the next two terms  1.8, 2.0, 2.2, 2.4, ...

Find the common difference of the A.P. and write the next two terms  \[0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, . . . \]

Find the common difference of the A.P. and write the next two terms  119, 136, 153, 170, ... 

Find the 10th term of the AP 1,4, 7, 10….

Find the 18^th term of the AP `sqrt2, 3sqrt2, 5sqrt2.....`

Find the n^th term of the A.P. 13, 8, 3, −2, ...

Find the 10^th term of the A.P. −40, −15, 10, 35, ...

Find 8th term of the A.P. 117, 104, 91, 78, ...

Find 11th term of the A.P. 10.0, 10.5, 11.0, 11.5, ...

Find 9th term of the A.P `3/4, 5/4, 7/4, 9/4,.....`

Which term of the A.P. 3, 8, 13, ... is 248?

Which term of the A.P. 84, 80, 76, ... is 248?

Which term of the A.P. 4, 9, 14, ... is 254?

Which term of the A.P. 21, 42, 63, 84, ... is 420?

Which term of the A.P. 121, 117, 113 … is its first negative term?

[Hint: Find n for a_n < 0]

Find Is 68 a term of the A.P. 7, 10, 13, ...?

Is 302 a term of the A.P. 3, 8, 13, ...?

Is −150 a term of the A.P. 11, 8, 5, 2, ...?

How many terms are there in the A.P.?

7, 10, 13, ... 43.

How many terms are there in the A.P.?

`-1, 5/6, 2/3, 1/2,.....10/3`

Find the number of terms in the following A.P.: 7, 13, 19, ..., 205.

Find the number of terms in the following A.P.

`18,15 1/2, 13`, ..., – 47

The first term of an A.P. is 5, the common difference is 3 and the last term is 80;

The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.

If the 9^th term of an A.P. is zero, then prove that 29^th term is double of 19^th term.

If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.

The 10^th and 18^th terms of an A.P. are 41 and 73 respectively. Find 26^th term.

In a certain A.P. the 24^th term is twice the 10^th term. Prove that the 72nd term is twice the 34th term.

If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.

Find the 12^th term from the end of the following arithmetic progressions:

3, 5, 7, 9, ... 201

Find the 12^th term from the end of the following arithmetic progressions:

3, 8, 13, ..., 253

Find the 12^th term from the end of the following arithmetic progressions:

1, 4, 7, 10, ..., 88

The 4^th term of an A.P. is three times the first and the 7^th term exceeds twice the third term by 1. Find the first term and the common difference.

Find the second term and n^th term of an A.P. whose 6^th term is 12 and 8^th term is 22.

How many numbers of two digit are divisible by 3?

An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find the 32nd term.

The sum of 4^th and 8^th terms of an A.P. is 24 and the sum of the 6^th and 10^th terms is 34. Find the first term and the common difference of the A.P.

The first term of an A.P. is 5 and its 100^th term is -292. Find the 50^th term of this A.P.

Find a₃₀ − a₂₀ for the A.P.

−9, −14, −19, −24, ...

Find a₃₀ − a₂₀ for the A.P.

a,a + d, a + 2d, a + 3d, ...

Write the expression a_n- a_k for the A.P. a, a + d, a + 2d, ... Hence, find the common difference of the A.P. for which

11^th term is 5 and the 13^th term is 79.

Write the expression a_n- a_k for the A.P. a, a + d, a + 2d, ... Hence, find the common difference of the A.P. for which  a₁₀ −a₅ = 200

Write the expression a_n- a_k for the A.P. a, a + d, a + 2d, ... Hence, find the common difference of the A.P. for which

Find n if the given value of x is the nth term of the given A.P.

25, 50, 75, 100, ...; x = 1000

Find n if the given value of x is the nth term of the given A.P.

−1, −3, −5, −7, ...; x = −151

Find n if the given value of x is the nth term of the given A.P.

`5 1/2, 11, 16 1/2, 22, ......; x = 550`

Find n if the given value of x is the nth term of the given A.P.

`1, 21/11, 31/11, 41/11,......, x = 171/11`

Find the arithmetic progression whose third term is 16 and the seventh term exceeds its fifth term by 12.

The 7^th term of an A.P. is 32 and its 13^th term is 62.  Find the A.P.

Which term of the A.P. 3, 10, 17, ... will be 84 more than its 13th term?

Two arithmetic progressions have the same common difference. The difference between their 100th terms is 100, what is the difference between their l000th terms?

For what value of n, the n^th terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... equal?

How many multiples of 4 lie between 10 and 250?

Which term of the A.P. 8, 14, 20, 26, ... will be 72 more than its 41^st term?

Find the term of the arithmetic progression 9, 12, 15, 18, ... which is 39 more than its 36^th term.

Find the 8^th term from the end of the A.P. 7, 10, 13, ..., 184

Find the 10^th term from the end of the A.P. 8, 10, 12, ..., 126.

The sum of 4th and 8th terms of an A.P. is 24 and the sum of 6th and 10th terms is 44. Find the A.P.

Which term of the A.P. 3, 15, 27, 39, ... will be 120 more than its 21^st term?

The 17th term of an A.P. is 5 more than twice its 8th term. If the 11th term of the A.P. is 43, find the nth term.

The 19^th term of an A.P. is equal to three times its sixth term. If its 9^th term is 19, find the A.P.

The 9^th term of an A.P. is equal to 6 times its second term. If its 5^th term is 22, find the A.P.

The 24^th term of an A.P. is twice its 10^th term. Show that its 72^nd term is 4 times its 15^thterm.

Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

If the seventh term of an A.P. is  \[\frac{1}{9}\] and its ninth term is \[\frac{1}{7}\] , find its (63)^rd term.

The sum of 5^th and 9^th terms of an A.P. is 30. If its 25^th term is three times its 8^th term, find the A.P.

If (m + 1)^th term of an A.P is twice the (n + 1)^th term, prove that (3m + 1)^th term is twice the (m + n + 1)^th term.

If an A.P. consists of n terms with first term a and n^th term l show that the sum of the m^th term from the beginning and the m^th term from the end is (a + l).

Find the value of x for which (8x + 4), (6x − 2) and (2x + 7) are in A.P.

If x + 1, 3x and 4x + 2 are in A.P., find the value of x.

Show that (a − b)², (a² + b²) and (a + b)² are in A.P.

The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceed the second term by 6, find three terms.

Three numbers are in A.P. If the sum of these numbers is 27 and the product 648, find the numbers.

Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.

The sum of three numbers in A.P. is 12 and sum of their cubes is 288. Find the numbers.

Let the four terms of the AP be a − 3d, a − d, a + d and a + 3d. find A.P.

The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.

Suppose three parts of 207 are (a − d), a , (a + d) such that , (a + d)  >a >  (a − d). 

Suppose the angles of a triangle are (a − d), a , (a + d) such that , (a + d)  >a >  (a − d).

Find the sum of the following arithmetic progressions: 50, 46, 42, ... to 10 terms

Find the sum of the following arithmetic progressions:

1, 3, 5, 7, ... to 12 terms

Find the sum of the following arithmetic progressions:

3, 9/2, 6, 15/2, ... to 25 terms

Find the sum of the following arithmetic progressions:

41, 36, 31, ... to 12 terms

Find the sum of the following arithmetic progressions:

a + b, a − b, a − 3b, ... to 22 terms

Find the sum of the following arithmetic progressions 

`(x - y)^2,(x^2 + y^2), (x + y)^2,.... to n term`

Find the sum of the following arithmetic progressions:

`(x - y)/(x + y),(3x - 2y)/(x + y), (5x - 3y)/(x + y)`,  .....to n terms

Find the sum of the following arithmetic progressions:

−26, −24, −22, …. to 36 terms

Find the sum to n term of the A.P. 5, 2, −1, −4, −7, ...,

Find the sum of n terms of an A.P. whose nth terms is given by a_n = 5 − 6n.

The term  A.P is 8, 10, 12, 14,...., 126 . find A.P.

Find the sum of the first 15 terms of each of the following sequences having the nth term as

`a_n = 3 + 4n`

Find the sum of the first 15 terms of each of the following sequences having the nth term as

b_n = 5 + 2n

Find the sum of the first 15 terms of each of the following sequences having nth term as  x_n = 6 − n .

Find the sum of the first 15 terms of each of the following sequences having the nth term as

y_n = 9 − 5n

Find the sum of first 20 terms of the sequence whose nth term is `a_n = An + B`

Find the sum of the first 25 terms of an A.P. whose nth term is given by a_n = 2 − 3n.

Find the sum of the first 25 terms of an A.P. whose nth term is given by a_n = 7 − 3n

If the sum of a certain number of terms starting from first term of an A.P. is 25, 22, 19, ..., is 116. Find the last term.

How many terms of the A.P. 18, 16, 14, .... be taken so that their sum is zero?

How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?

How many terms of the AP. 9, 17, 25 … must be taken to give a sum of 636?

How many terms of the A.P. 63, 60, 57, ... must be taken so that their sum is 693?

Find the sum of the first 11 terms of the A.P : 2, 6, 10, 14, ...

Find the sum of the first 13 terms of the A.P: -6, 0, 6, 12,....

Find the sum of the first 51 terms of the A.P: whose second term is 2 and the fourth term is 8.

Find the sum of first 15 multiples of 8.

Find the sum of the first 40 positive integers divisible by 3

Find the sum of the first 40 positive integers divisible by 5

Find the sum of first 40 positive integers divisible by 6.

Find the sum of all 3 - digit natural numbers which are divisible by 13.

Find the sum of all 3-digit natural numbers, which are multiples of 11.

Find the sum of all 2 - digit natural numbers divisible by 4.

Find the sum 2 + 4 + 6 ... + 200

Find the sum 3 + 11 + 19 + ... + 803

Find the sum  (−5) + (−8)+ (−11) + ... + (−230) .

Find the sum:  1 + 3 + 5 + 7 + ... + 199 .

Find the sum \[7 + 10\frac{1}{2} + 14 + . . . + 84\]

Find the sum given below:

34 + 32 + 30 + ... + 10

Find the sum 25 + 28 + 31 + ….. + 100

The first and last terms of an AP are 17 and 350, respectively. If the common difference is 9, how many terms are there, and what is their sum?

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.

The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

If the 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?

Find the sum of n terms of the series \[\left( 4 - \frac{1}{n} \right) + \left( 4 - \frac{2}{n} \right) + \left( 4 - \frac{3}{n} \right) + . . . . . . . . . .\]

In an A.P., if the first term is 22, the common difference is −4 and the sum to n terms is 64,  find n.

In an A.P., if the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms?

Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.

The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

In an A.P. the first term is 8, nth term is 33 and the sum to first n terms is 123. Find n and d, the common differences.

In an A.P., the first term is 22, nth term is −11 and the sum to first n terms is 66. Find n and d, the common difference

​The first and the last terms of an A.P. are 7 and 49 respectively. If sum of all its terms is 420, find its common difference. 

The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

The sum of first 9 terms of an A.P. is 162. The ratio of its 6^th term to its 13^th term is 1 : 2. Find the first and 15^th term of the A.P.

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28^th term of this A.P.

The sum of first seven terms of an A.P. is 182. If its 4^th and the 17^th terms are in the ratio 1 : 5, find the A.P.

The nth term of an AP is given by (−4n + 15). Find the sum of first 20 terms of this AP?

In an A.P., the sum of first ten terms is −150 and the sum of its next ten terms is −550. Find the A.P.

Sum of the first 14 terms of and AP is 1505 and its first term is 10. Find its 25^th term. 

In an AP, the first term is 2, the last term is 29 and the sum of all the terms is 155. Find the common difference.

The first and last terms of an AP are 17 and 350, respectively. If the common difference is 9, how many terms are there, and what is their sum?

Find the number of terms of the AP − 12, −9, −6, ….., 12. If 1 is added to each term of this AP, then find the sum of all terms of the AP thus obtained ?

The sum of first q terms of an AP is (63q – 3q²⁾. If its p^th term is –60, find the value of p. Also, find the 11^th term of its AP.

The sum of fist m terms of an AP is ( 4m²  - m). If its nth term is 107, find the value of n. Also, Find the 21^st term of this AP. 

If the sum of the first n terms of an AP is 4n − n², what is the first term (that is, S₁)? What is the sum of the first two terms? What is the second term? Similarly, find the 3^rd, the 10th, and the n^th terms.

If the sum of first n terms of an A.P. is  \[\frac{1}{2}\] (3n² + 7n), then find its n^th term. Hence write its 20^th term.

In an A.P., the sum of first n terms is `(3n^2)/2 + 13/2 n`. Find its 25^th term.

Find the sum of all natural numbers between 1 and 100, which are divisible by 3.

Find the sum of first n odd natural numbers

Find the sum of the odd numbers between 0 and 50.

Find the sum of all odd numbers between 100 and 200.

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

Find the sum of all integers between 84 and 719, which are multiples of 5.

Find the sum of all integers between 50 and 500, which are divisible by 7.

Find the sum of all even integers between 101 and 999.

Find the sum of all integers between 100 and 550, which are divisible by 9.

In an AP: Given a = 5, d = 3, a_n = 50, find n and S_n.

In an AP given a_n = 4, d = 2, S_n = −14, find n and a.

 In an AP given a = 3, n = 8, S_n = 192, find d.

In an AP given l = 28, S = 144, and there are total 9 terms. Find a.

In an AP given a = 8, a_n = 62, S_n = 210, find n and d.

In an AP, given a = 2, d = 8, and S_n = 90, find n and a_n.

Let there be an A.P. with first term 'a', common difference 'd'. If a_n denotes in n^th term and S_n the sum of first n terms, find.

Find the sum of first 22 terms of an A.P. in which d = 22 and a = 149.

In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students.

Ramkali would need ₹1800 for admission fee and books etc., for her daughter to start going to school from next year. She saved ₹50 in the first month of this year and increased her monthly saving by ₹20. After a year, how much money will she save? Will she be able to fulfil her dream of sending her daughter to school?

A man saved Rs 16500 in ten years. In each year after the first, he saved Rs 100 more than he did in the preceding year. How much did he save in the first year?

A man saved Rs. 32 during the first year, Rs 36 in the second year and in this way he increases his saving by Rs 4 every year. Find in what time his saving will be Rs. 200.

A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of all debt unpaid, finds the value of the first instalment.

There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.

A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.

A piece of equipment cost a certain factory Rs 60,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?

A sum of Rs 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.

x is nth term of the given A.P. a_n = x find x .

Write the expression of the common difference of an A.P. whose first term is a and nth term is b.

If `4/5` , a, 2 are three consecutive terms of an A.P., then find the value of a. 

The given terms are 2k + 1, 3k + 3 and 5k − 1. find AP.

Write the nth term of the \[A . P . \frac{1}{m}, \frac{1 + m}{m}, \frac{1 + 2m}{m}, . . . .\]

Mark the correct alternative in each of the following:
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

If the sum of P terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be

If the sum of n terms of an A.P. be 3n² + n and its common difference is 6, then its first term is ______.

The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be

If the sum of n terms of an A.P. is 3n² + 5n then which of its terms is 164?

If the sum of n terms of an A.P. is 2n² + 5n, then its nth term is

If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times, the least, then the numbers are

Let S_n denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = S_n − kS_n−1 + S_n−2, then k =

The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] , then k =

If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =

If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is

If S₁ is the sum of an arithmetic progression of 'n' odd number of terms and S₂ the sum of the terms of the series in odd places, then \[\frac{S_1}{S_2} =\]

If in an A.P. S_n = n²p and S_m = m²p, where S_r denotes the sum of r terms of the A.P., then S_p is equal to 

If S_n denote the sum of the first n terms of an A.P. If S_2n = 3S_n, then S_3n : S_n is equal to

In an AP. S_p = q, S_q = p and S_r denotes the sum of first r terms. Then, S_p+q is equal to

If S_r denotes the sum of the first r terms of an A.P. Then , S_3n: (S_2n − S_n) is

If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 terms is 

The number of terms of the A.P. 3, 7, 11, 15, ... to be taken so that the sum is 406 is

Sum of n terms of the series  `sqrt2+sqrt8+sqrt18+sqrt32+....` is ______.

The 9th term of an A.P. is 449 and 449th term is 9. The term which is equal to zero is

If \[\frac{1}{x + 2}, \frac{1}{x + 3}, \frac{1}{x + 5}\]  are in A.P. Then, x =

The nth term of an A.P., the sum of whose n terms is S_n, is

The common difference of an A.P., the sum of whose n terms is S_n, is

If the sums of n terms of two arithmetic progressions are in the ratio \[\frac{3n + 5}{5n - 7}\] , then their n^th terms are in the ratio

If S_n denote the sum of n terms of an A.P. with first term a and common difference dsuch that \[\frac{Sx}{Skx}\]  is independent of x, then

If the first term of an A.P. is a and nth term is b, then its common difference is

The sum of first n odd natural numbers is ______.

Two A.P.'s have the same common difference. The first term of one of these is 8 and that of the other is 3. The difference between their 30th term is

If 18, a, b, −3 are in A.P., the a + b =

The sum of n terms of two A.P.'s are in the ratio 5n + 9 : 9n + 6. Then, the ratio of their 18th term is

If \[\frac{5 + 9 + 13 + . . . \text{ to n terms} }{7 + 9 + 11 + . . . \text{ to (n + 1) terms}} = \frac{17}{16},\] then n =

The sum of n terms of an A.P. is 3n² + 5n, then 164 is its

If the nth term of an A.P. is 2n + 1, then the sum of first n terms of the A.P. is 

If 18^th and 11^th term of an A.P. are in the ratio 3 : 2, then its 21^st and 5^th terms are in the ratio

The sum of first 20 odd natural numbers is

The common difference of the A.P. is \[\frac{1}{2q}, \frac{1 - 2q}{2q}, \frac{1 - 4q}{2q}, . . .\] is 

The common difference of the A.P.

The common difference of the A.P. \[\frac{1}{2b}, \frac{1 - 6b}{2b}, \frac{1 - 12b}{2b}, . . .\] is 

If k, 2k − 1 and 2k + 1 are three consecutive terms of an A.P., the value of k is

The next term of the A.P. \[\sqrt{7}, \sqrt{28}, \sqrt{63}\] is ______.

The first three terms of an A.P. respectively are 3y − 1, 3y + 5 and 5y + 1. Then, y equals