RD Sharma solutions for Mathematics [English] Class 11 chapter 19 - Arithmetic Progression [Latest edition]

A sequence is defined by a_n = n³ − 6n² + 11n − 6, n ϵ N. Show that the first three terms of the sequence are zero and all other terms are positive.

Let < a_n > be a sequence defined by a₁ = 3 and, a_n = 3a_{n − 1} + 2, for all n > 1
Find the first four terms of the sequence.

Let < a_n > be a sequence. Write the first five term in the following:

a₁ = 1, a_n = a_{n − 1} + 2, n ≥ 2

Let < a_n > be a sequence. Write the first five term in the following:

a₁ = 1 = a₂, a_n = a_{n − 1} + a_{n − 2}, n > 2

Let < a_n > be a sequence. Write the first five term in the following:

a₁ = a₂ = 2, a_n = a_{n − 1} − 1, n > 2

The Fibonacci sequence is defined by a₁ = 1 = a₂, a_n = a_{n − 1} + a_{n − 2} for n > 2

Find `(""^an +1)/(""^an")` for n = 1, 2, 3, 4, 5.

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.

 3, −1, −5, −9 ...

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.

−1, 1/4, 3/2, 11/4, ...

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case.

\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2}, 7\sqrt{2}, . . .\]

Show that the following sequence is an A.P. Also find the common difference and write 3 more terms in case. 

9, 7, 5, 3, ...

The n^th term of a sequence is given by a_n = 2n² + n + 1. Show that it is not an A.P.

Find:

 10th term of the A.P. 1, 4, 7, 10, ...

Find: 

18th term of the A.P.

\[\sqrt{2}, 3\sqrt{2}, 5\sqrt{2},\]

Find:

nth term of the A.P. 13, 8, 3, −2, ...

If the sequence < a_n > is an A.P., show that a_m +n +a_m − n = 2a_m.

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

Which term of the A.P. 84, 80, 76, ... is 0?

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

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

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

Which term of the sequence 24, \[23\frac{1}{4,} 22\frac{1}{2,} 21\frac{3}{4}\]....... is the first negative term?

Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely real ?

Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?

How many terms are there in the A.P. 7, 10, 13, ... 43 ?

How many terms are there in the A.P.\[- 1, - \frac{5}{6}, -\frac{2}{3}, - \frac{1}{2}, . . . , \frac{10}{3}?\] 

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

The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th 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 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.

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

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.

Find the 12th term from the following arithmetic progression:

 3, 5, 7, 9, ... 201

Find the 12th term from the following arithmetic progression:

3, 8, 13, ..., 253

Find the 12th term from the following arithmetic progression:

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

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

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

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

How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?

The first and the last terms of an A.P. are a and l respectively. Show that the sum of nthterm from the beginning and nth term from the end is a + l.

If < a_n > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].

\[\text { If } \theta_1 , \theta_2 , \theta_3 , . . . , \theta_n \text { are in AP, whose common difference is d, then show that }\]

\[\sec \theta_1 \sec \theta_2 + \sec \theta_2 \sec \theta_3 + . . . + \sec \theta_{n - 1} \sec \theta_n = \frac{\tan \theta_n - \tan \theta_1}{\sin d} \left[ NCERT \hspace{0.167em} EXEMPLAR \right]\]

Three numbers are in A.P. If the sum of these numbers be 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 the sum of their cubes is 288. Find the numbers.

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

Find the sum of the following arithmetic progression :

50, 46, 42, ... to 10 terms

Find the sum of the following arithmetic progression :

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

Find the sum of the following arithmetic progression :

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

Find the sum of the following arithmetic progression :

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

Find the sum of the following arithmetic progression :

a + b, a − b, a − 3b, ... to 22 terms

Find the sum of the following arithmetic progression :

 (x − y)², (x² + y²), (x + y)², ... to n terms

Find the sum of the following arithmetic progression :

\[\frac{x - y}{x + y}, \frac{3x - 2y}{x + y}, \frac{5x - 3y}{x + y}\], ... to n terms.

Find the sum of the following serie:

 2 + 5 + 8 + ... + 182

Find the sum of the following serie:

101 + 99 + 97 + ... + 47

Find the sum of the following serie:

(a − b)² + (a² + b²) + (a + b)² + ... + [(a + b)² + 6ab]

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 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.

Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.

Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.

Solve: 

25 + 22 + 19 + 16 + ... + x = 115

Solve: 

1 + 4 + 7 + 10 + ... + x = 590.

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?

The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.

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.

The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by \[10 \frac{1}{2}\] ,find the number of terms and the series. 

If 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?

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

How many terms of the A.P. −6, \[- \frac{11}{2}\], −5, ... are needed to give the sum −25?

If S₁ be the sum of (2n + 1) terms of an A.P. and S₂ be the sum of its odd terms, then prove that: S₁ : S₂ = (2n + 1) : (n + 1).

Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.

If the sum of n terms of an A.P. is nP + \[\frac{1}{2}\] n (n − 1) Q, where P and Q are constants, find the common difference.

The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.

The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.

If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:

\[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P.

If \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P., prove that:

a (b +c), b (c + a), c (a +b) are in A.P.

If a², b², c² are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.

If a, b, c is in A.P., then show that:

 a² (b + c), b² (c + a), c² (a + b) are also in A.P.

If a, b, c is in A.P., then show that:

b + c − a, c + a − b, a + b − c are in A.P.

If a, b, c is in A.P., then show that:

bc − a², ca − b², ab − c² are in A.P.

If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:

\[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.

If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:

 bc, ca, ab are in A.P.

If a, b, c is in A.P., prove that:

 (a − c)² = 4 (a − b) (b − c)

If a, b, c is in A.P., prove that:

a² + c² + 4ac = 2 (ab + bc + ca)

If a, b, c is in A.P., prove that:

 a³ + c³ + 6abc = 8b³.

If \[a\left( \frac{1}{b} + \frac{1}{c} \right), b\left( \frac{1}{c} + \frac{1}{a} \right), c\left( \frac{1}{a} + \frac{1}{b} \right)\] are in A.P., prove that a, b, c are in A.P.

Show that x² + xy + y², z² + zx + x² and y² + yz + z² are consecutive terms of an A.P., if x, y and z are in A.P. 

Find the A.M. between:

 7 and 13 

Find the A.M. between:

12 and −8

Find the A.M. between:

(x − y) and (x + y).

Insert 4 A.M.s between 4 and 19.

Insert 7 A.M.s between 2 and 17.

Insert six A.M.s between 15 and −13.

If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.

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

A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings 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 the debt unpaid, find the value of the first instalment.

A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.

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 600,000. 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?

Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.

The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.

A man starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?

A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job? 

We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.

In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?

A man accepts a position with an initial salary of ₹5200 per month. It is understood that he will receive an automatic increase of ₹320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?

A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?

In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?

Write the common difference of an A.P. whose nth term is xn + y.

Write the common difference of an A.P. the sum of whose first n terms is

If the sum of n terms of an AP is 2n² + 3n, then write its nth term.

If log 2, log (2^x − 1) and log (2^x + 3) are in A.P., write the value of x.

If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.

Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.

If \[\frac{3 + 5 + 7 + . . . + \text { upto n terms }}{5 + 8 + 11 + . . . . \text { upto 10 terms }}\] 7, then find the value of n.

If m th term of an A.P. is n and nth term is m, then write its pth term.

If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.

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 3 n² − n and its common difference is 6, then its first term is

Sum of all two digit numbers which when divided by 4 yield unity as remainder is

In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is

If S_n denotes the sum of first n terms of an A.P. < a_n > such that

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 3 n² + 5 n then which of its terms is 164?

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

If a₁, a₂, a₃, .... a_n are in A.P. with common difference d, then the sum of the series sin d [cosec a₁cosec a₂ + cosec a₁ cosec a₃ + .... + cosec a_{n − 1} cosec a_n] is

In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is

If a₁, a₂, a₃, .... a_n are in A.P. with common difference d, then the sum of the series sin d [sec a₁ sec a₂ + sec a₂ sec a₃ + .... + sec a_{n − 1} sec a_n], 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

If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is

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 − k S_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, then 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

Mark the correct alternative in the following question:
If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is

Mark the correct alternative in the following question:
The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is

Mark the correct alternative in the following question:

\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P  . , then }S_q \text { equals }\]

Mark the correct alternative in the following question:

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