Samacheer Kalvi solutions for Mathematics [English] Class 10 SSLC TN Board chapter 2 - Numbers and Sequences [Latest edition]

Find all positive integers, when divided by 3 leaves remainder 2

A man has 532 flower pots. He wants to arrange them in rows such that each row contains 21 flower pots. Find the number of completed rows and how many flower pots are left over

Prove that the product of two consecutive positive integers is divisible by 2

When the positive integers a, b and c are divided by 13, the respective remainders are 9, 7 and 10. Show that a + b + c is divisible by 13

Prove that square of any integer leaves the remainder either 0 or 1 when divided by 4

Use Euclid’s Division Algorithm to find the Highest Common Factor (H.C.F) of 340 and 412

Use Euclid’s Division Algorithm to find the Highest Common Factor (H.C.F) of 867 and 255

Use Euclid’s Division Algorithm to find the Highest Common Factor (H.C.F) of 10224 and 9648

Use Euclid’s Division Algorithm to find the Highest Common Factor (H.C.F) of 84, 90 and 120

Find the largest number which divides 1230 and 1926 leaving remainder 12 in each case

If d is the Highest Common Factor of 32 and 60, find x and y satisfying d = 32x + 60y

A positive integer, when divided by 88, gives the remainder 61. What will be the remainder when the same number is divided by 11?

Prove that two consecutive positive integers are always co-prime

For what value of natural number n, 4ⁿ can end with the digit 6?

If m, n are natural numbers, for what values of m, does 2ⁿ × 5^m ends in 5?

Find the H.C.F. of 252525 and 363636

If 13824 = 2^a × 3^b then find a and b

If p₁^x₁ × p₂^x₂ × p₃^x₃ × p₄^x₄ = 113400 where p₁, p₂, p₃, p₄ are primes in ascending order and x₁, x₂, x₃, x₄, are integers, find the value of p₁, p₂, p₃, p₄ and x₁, x₂, x₃, x₄

Find the L.C.M. and H.C.F. of 408 and 170 by applying the fundamental theorem of Arithmetic

Find the greatest number consisting of 6 digits which is exactly divisible by 24, 15, 36?

What is the smallest number that when divided by three numbers such as 35, 56 and 91 leaves remainder 7 in each case?

Find the least number that is divisible by the first ten natural numbers

Find the least positive value of x such that 71 ≡ x (mod 8)

Find the least positive value of x such that 78 + x ≡  3 (mod 5)

Find the least positive value of x such that 89 ≡ (x + 3) (mod 4)

Find the least positive value of x such that 96 ≡ `x/7` (mod 5)

Find the least positive value of x such that 5x ≡ 4 (mod 6)

If x is congruent to 13 modulo 17 then 7x – 3 is congruent to which number modulo 17?

Solve 5x ≡ 4 (mod 6)

Solve 3x – 2 ≡ 0 (mod 11)

What is the time 100 hours after 7 a.m.?

What is time 15 hours before 11 p.m.?

Today is Tuesday. My uncle will come after 45 days. In which day my uncle will be coming?

Prove that 2ⁿ + 6 × 9ⁿ is always divisible by 7 for any positive integer n

Find the remainder when 2⁸¹ is divided by 17.

The duration of flight travel from Chennai to London through British Airlines is approximately 11 hours. The airplane begin its journey on Sunday at 23:30 hours. If the time at Chennai is four and half hours ahead to that of London’s time, then find the time at London, when will the flight land at London Airport?

Find the next three terms of the following sequence.

8, 24, 72, …

Find the next three terms of the following sequence.

5, 1, – 3, …

Find the next three terms of the following sequence

`1/4, 2/9, 3/16, ...`

Find the first four terms of the sequence whose n^th terms are given by a_n = n³ – 2

Find the first four terms of the sequence whose n^th terms are given by a_n = (–1)ⁿ⁺¹ n(n + 1)

Find the first four terms of the sequences whose n^th terms are given by a_n = 2n² – 6

Find the n^th term of the following sequence

2, 5, 10, 17, ...

Find the n^th term of the following sequence

`0, 1/2, 2/3, ...`

Find the n^th term of the following sequence

3, 8, 13, 18, ...

Find the indicated terms of the sequences whose n^th terms are given by

a_n = `(5"n")/("n" + 2)`; a₆ and a₁₃

Find the indicated terms of the sequences whose n^th terms are given by

a_n = – (n² – 4); a₄ and a₁₁

Find a₈ and a₁₅ whose n^th term is a_n = `{{:(("n"^2 - 1)/("n" + 3)";", "n is even"","  "n ∈ N"),(("n"^2)/(2"n" + 1)";", "n is odd"","  "n ∈ N"):}`

If a₁ = 1, a₂ = 1 and a_n = 2a_n−1 + a_n−2, n ≥ 3, n ∈ N, then find the first six terms of the sequence

Check whether the following sequence is in A.P.

a – 3, a – 5, a – 7, …

Check whether the following sequence is in A.P.

`1/2, 1/3, 1/4, 1/5, ...`

Check whether the following sequence is in A.P.

9, 13, 17, 21, 25, ...

Check whether the following sequence is in A.P.

`(-1)/3, 0, 1/3, 2/3, ...`

Check whether the following sequence is in A.P.

1, –1, 1, –1, 1, –1, …

First term a and common difference d are given below. Find the corresponding A.P.

a = 5, d = 6

First term a and common difference d are given below. Find the corresponding A.P.

a = 7, d = – 5

First term a and common difference d are given below. Find the corresponding A.P.

a = `3/4`, d = `1/2`

Find the first term and common difference of the Arithmetic Progressions whose n^th term is given below

t_n = – 3 + 2n

Find the first term and common difference of the Arithmetic Progressions whose n^th term is given below

t_n = 4 – 7n

Find the 19^th term of an A.P. – 11, – 15, – 19, ...

Which term of an A.P. 16, 11, 6, 1, ... is – 54?

Find the middle term(s) of an A.P. 9, 15, 21, 27, …, 183.

If nine times ninth term is equal to the fifteen times fifteenth term, show that six times twenty fourth term is zero

If 3 + k, 18 – k, 5k + 1 are in A.P. then find k

Find x, y and z, given that the numbers x, 10, y, 24, z are in A.P.

In a theatre, there are 20 seats in the front row and 30 rows were allotted. Each successive row contains two additional seats than its front row. How many seats are there in the last row?

The sum of three consecutive terms that are in A.P. is 27 and their product is 288. Find the three terms

The ratio of 6^th and 8^th term of an A.P. is 7:9. Find the ratio of 9^th term to 13^th term

In the winter season let us take the temperature of Ooty from Monday to Friday to be in A.P. The sum of temperatures from Monday to Wednesday is 0°C and the sum of the temperatures from Wednesday to Friday is 18°C. Find the temperature on each of the five days.

Priya earned ₹ 15,000 in the first month. Thereafter her salary increased by ₹ 1500 per year. Her expenses are ₹ 13,000 during the first year and the expenses increase by ₹ 900 per year. How long will it take for her to save ₹ 20,000 per month

Find the sum of the following

3, 7, 11, … up to 40 terms

Find the sum of the following

102, 97, 92, … up to 27 terms.

Find the sum of the following

6 + 13 + 20 + .... + 97

How many consecutive odd integers beginning with 5 will sum to 480?

Find the sum of first 28 terms of an A.P. whose n^th term is 4n – 3

The sum of first n terms of a certain series is given as 2n² – 3n. Show that the series is an A.P.

The 104^th term and 4^th term of an A.P. are 125 and 0. Find the sum of first 35 terms

Find the sum of all odd positive integers less than 450

Find the sum of all natural numbers between 602 and 902 which are not divisible by 4.

Raghu wish to buy a laptop. He can buy it by paying ₹ 40,000 cash or by giving it in 10 installments as ₹ 4800 in the first month, ₹ 4750 in the second month, ₹ 4700 in the third month and so on. If he pays the money in this fashion, find total amount paid in 10 installments

Raghu wish to buy a laptop. He can buy it by paying ₹ 40,000 cash or by giving it in 10 installments as ₹ 4800 in the first month, ₹ 4750 in the second month, ₹ 4700 in the third month and so on. If he pays the money in this fashion, find how much extra amount that he has to pay than the cost?

A man repays a loan of ₹ 65,000 by paying ₹ 400 in the first month and then increasing the payment by ₹ 300 every month. How long will it take for him to clear the loan?

A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two bricks less than the previous step

How many bricks are required for the top most step?

A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two bricks less than the previous step

How many bricks are required to build the stair case?

If S₁, S₂ , S₃, …., S_m are the sums of n terms of m A.P.'s whose first terms are 1, 2, 3, ……, m and whose common differences are 1, 3, 5, …., (2m – 1) respectively, then show that S₁ + S₂ + S₃ + ... + S_m = `1/2` mn(mn + 1)

Find the sum `[("a" - "b")/("a" + "b") + (3"a" - 2"b")/("a" +  "b") + (5"a" - 3"b")/("a" + "b") + ...  "to"  12  "terms"]`

Identify the following sequence is in G.P.?

3, 9, 27, 81, …

Identify the following sequence is in G.P.?

4, 44, 444, 4444, .....

Identify the following sequence is in G.P.?

0.5, 0.05, 0.005, …

Identify the following sequence is in G.P.?

`1/3, 1/6, 1/12, ...`

Identify the following sequence is in G.P.?

1, −5, 25, −125, ...

Identify the following sequence is in G.P.?

120, 60, 30, 18, …

Identify the following sequence is in G.P.?

`16, 4, 1, 1/4,...`

Write the first three terms of the G.P. whose first term and the common ratio are given below

a = 6, r = 3

Write the first three terms of the G.P. whose first term and the common ratio are given below

a = `sqrt(2)`, r = `sqrt(2)`

Write the first three terms of the G.P. whose first term and the common ratio are given below

a = 1000, r = `2/5`

In a G.P. 729, 243, 81, … find t₇

Find x so that x + 6, x + 12 and x + 15 are consecutive terms of a Geometric Progression

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

4, 8, 16, …, 8192?

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

`1/3, 1/9, 1/27, ..., 1/2187`

In a G.P. the 9^th term is 32805 and 6^th term is 1215. Find the 12^th term

Find the 10^th term of a G.P. whose 8^th term is 768 and the common ratio is 2

If a, b, c is in A.P. then show that 3^a, 3^b, 3^c  is in G.P.

In a G.P. the product of three consecutive terms is 27 and the sum of the product of two terms taken at a time is `57/2`. Find the three terms

A man joined a company as Assistant Manager. The company gave him a starting salary of ₹ 60,000 and agreed to increase his salary 5% annually. What will be his salary after 5 years?

Sivamani is attending an interview for a job and the company gave two offers to him. Offer A: ₹ 20,000 to start with followed by a guaranteed annual increase of 6% for the first 5 years. Offer B: ₹ 22,000 to start with followed by a guaranteed annual increase of 3% for the first 5 years

If a, b, c is three consecutive terms of an A.P. and x, y, z are three consecutive terms of a G.P. then prove that x^b–c × y^c–a × z^a–b = 1

Find the sum of first n terms of the G.P.

`5, -3, 9/5, - 27/25, ...`

Find the sum of first n terms of the G.P.

256, 64, 16, …

Find the sum of first six terms of the G.P. 5, 15, 45, …

Find the first term of the G.P. whose common ratio 5 and whose sum to first 6 terms is 46872

Find the sum to infinity of 9 + 3 + 1 + ….

Find the sum to infinity of `21 + 14 + 28/3 + ...`

If the first term of an infinite G.P. is 8 and its sum to infinity is `32/3` then find the common ratio

Find the sum to n terms of the series

0.4 + 0.44 + 0.444 + …… to n terms

Find the sum to n terms of the series

3 + 33 + 333 + ………… to n terms

Find the sum of the Geometric series 3 + 6 + 12 + …….. + 1536

Kumar writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with the instruction that they continue the process similarly. Assuming that the process is unaltered and it costs ₹ 2 to mail one letter, find the amount spent on postage when 8^th set of letters is mailed

Find the rational form of the number `0.bar(123)`

If S_n = (x + y) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ... n terms then prove that (x – y)S_n = `[(x^2(x^"n" - 1))/(x - 1) - (y^2(y^"n" - 1))/(y - 1)]`

Find the sum of the following series

1 + 2 + 3 + ... + 60

Find the sum of the following series

3 + 6 + 9 + ... + 96

Find the sum of the following series

51 + 52 + 53 + ... + 92

Find the sum of the following series

1 + 4 + 9 + 16 + ... + 225

Find the sum of the following series

6² + 7² + 8² + ... + 21²

Find the sum of the following series

10³ + 11³ + 12³ + ... + 20³

Find the sum of the following series

1 + 3 + 5 + ... + 71

If 1 + 2 + 3 + ... + k = 325, then find 1³ + 2³ + 3³ + ... + k³ 

If 1³ + 2³ + 3³ + ... + k³ = 44100 then find 1 + 2 + 3 + ... + k

How many terms of the series 1³ + 2³ + 3³ + … should be taken to get the sum 14400?

The sum of the cubes of the first n natural numbers is 2025, then find the value of n

Rekha has 15 square colour papers of sizes 10 cm,11 cm,12 cm, …, 24 cm. How much area can be decorated with these colour papers?

Find the sum of the series (2³ – 1³) + (4³ – 3³) + (6³ – 15³) + ... to n terms

Find the sum of the series (2³ – 1³) + (4³ – 3³) + (6³ – 15³) + ... to 8 terms

Euclid’s division lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy

Using Euclid’s division lemma, if the cube of any positive integer is divided by 9 then the possible remainders are

If the H.C.F of 65 and 117 is expressible in the form of 65m – 117, then the value of m is

The sum of the exponents of the prime factors in the prime factorization of 1729 is

The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is

7^4k ≡ _____ (mod 100)

Given F₁ = 1, F₂ = 3 and F_n = F_n–1 + F_n–2 then F₅ is

The first term of an arithmetic progression is unity and the common difference is 4. Which of the following will be a term of this A.P.

If 6 times of 6^th term of an A.P. is equal to 7 times the 7^th term, then the 13^th term of the A.P. is

An A.P. consists of 31 terms. If its 16^th term is m, then the sum of all the terms of this A.P. is

In an A.P., the first term is 1 and the common difference is 4. How many terms of the A.P. must be taken for their sum to be equal to 120?

If A = 2⁶⁵ and B = 2⁶⁴ + 2⁶³ + 2⁶² …. + 2⁰ identify of the following is true?

The next term of the sequence `3/16, 1/8, 1/12, 1/18, ...` is

If the sequence t₁, t₂, t₃ … is in A.P. then the sequence t₆, t₁₂, t₁₈ … is

The value of (1³ + 2³ + 3³ + ... + 15³) – (1 + 2 + 3 + ... + 15) is

Prove that n² – n divisible by 2 for every positive integer n

A milk man has 175 litres of cow’s milk and 105 litres of buffalow’s milk. He wishes to sell the milk by filling the two types of milk in cans of equal capacity. Calculate the following :

Capacity of a can

A milk man has 175 litres of cow’s milk and 105 litres of buffalow’s milk. He wishes to sell the milk by filling the two types of milk in cans of equal capacity. Calculate the following :

Number of cans of cow’s milk

A milk man has 175 litres of cow’s milk and 105 litres of buffalow’s milk. He wishes to sell the milk by filling the two types of milk in cans of equal capacity. Calculate the following :

Number of cans of buffalow’s milk

When the positive integers a, b and c are divided by 13 the respective remainders is 9, 7 and 10. Find the remainder when a b + + 2 3c is divided by 13

Show that 107 is of the form  4q +3 for any integer q

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

Find the 12^th term from the last term of the A.P – 2, – 4, – 6, … – 100

Two A.P.’s have the same common difference. The first term of one A.P. is 2 and that of the other is 7. Show that the difference between their 10^th terms is the same as the difference between their 21^st terms, which is the same as the difference between any two corresponding terms.

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

Find the G.P. in which the 2^nd term is `sqrt(6)` and the 6^th term is `9sqrt(6)`

The value of a motorcycle depreciates at a rate of 15% per year. What will be the value of the motorcycle for 3 year hence, which is now purchased for ₹ 45,000?