Nootan solutions for Mathematics [English] Class 10 ICSE chapter 9 - Arithmetic and geometric progression [Latest edition]

For the given A.P.s, write the first term a and common difference d.

3, 5, 7, ....

For the given A.P.s, write the first term a and common difference d.

4, −1, −6, ....

For the given A.P.s, write the first term a and common difference d.

1.7, 2.3, 2.9, ...

Write the first four terms of the A.P. when the first term a and the common difference d are given.

a = 5, d = 2

Write the first four terms of the A.P. when the first term a and the common difference d are given.

a = 3, d = 0

Write the first four terms of the A.P. when the first term a and the common difference d are given.

a = −4, d = −1

Write the first four terms of the A.P. when the first term a and the common difference d are given.

a = `1/2`, d = `3/2`

The following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms.

1²,2², 3², 4²,...

The following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms.

3, 10, 17, 24,...

The following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms.

3, 0, −3, −6,...

The following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms.

3, 3, 5, 5,....

If 2k, 3k + 1, and 5k − 1 are three consecutive terms of an A.P., find the value of k.

If 11, a, b, 2, are in A.P., find the values of a and b.

The nth term of a progression is (3n + 5). Prove that this progression is an arithmetic progression. Also, find its 6th term.

The nth term of a progression is (3 – 4n). Prove that this progression is an arithmetic progression. Also, find its common difference.

The nth term of a progression is (n² − n + 1). Prove that it is not an A.P.

Find the 10th term of the progression 1 + 3 + 5 + 7 + ...

Find the 7th term of the progression 80 + 77 + 74 + ...

Find the 22nd term of the progression `7 3/4 + 9 1/2 + 11 1/4 + ...`

Find the nth term of the progression − 5 − 3 − 1 + 1 + ...

Which term of the progression 4 + 8 + 12 + ... is 76?

Which term of the progression 36 + 33 + 30 + ... is zero?

Which term of the progression `3/4 + 1 + 5/4 + ...` is 12?

Find the 16th term from the end of the progression 3 + 6 + 9 + ... + 99.

Find the 10th term from the end of the progression
82 + 79 + 76 + … + 4.

Find the 10th term from the end of the progression 5 + 2 − 1 − 4 − ... − 34.

How many numbers of two digit are divisible by 3?

How many numbers of three digits are divisible by 9?

Find the value of ‘x’ if x + 1, 2x + 1, and x + 7 are in A.P. Also, find the 4th term of this progression.

If k + 3, 2k + 1, k + 7 are in A.P., then find this progression up to 5 terms.

The 3rd and 19th terms of an A.P. are 13 and 77, respectively. Find the A.P.

The 5th and 8th terms of an A.P. are 56 and 95, respectively. Find the 25th term of this A.P.

The pth and qth terms of an A.P. are q and p, respectively. Prove that its (p + q)th terms will be zero.

If (p + 1) th term of an A.P. is twice the (q + 1)th term, then prove that (3p + 1)th term willbe twice the (p + q + 1)th term.

The 12th term of an A.P. is 14 more than the 5th term. The sum of the first three terms is 36. Find the A.P.

Is 303, a term of the progression 5, 10, 15, ...?

Is 38, a term of the progression −18, −14, −10, ....?

Prove that the sum of nth term from the beginning and nth term from the end of an A.P. is constant.

In an A.P., prove that: `T_(m + n ) + T_(m - n) = 2.T_m`

10 times the 10th term and 15 times the 15th term of an A.P. are equal. Find the 25th term of this A.Р.

17 times the 17th term of an A.P. is equal to 18 times the 18th term. Find the 35th term of this progression.

Which term of the progression `10, 9 1/3, 8 2/3,...` is the first negative term?

Which term of the progression `4, 3 5/7, 3 3/7,` is the first negative term?

Each of two arithmetic progressions 2, 4, 6, ... and 3, 6, 9, ... is taken up to 200 terms. How many terms are common in these two progressions?

Find three numbers in A.P. whose sum is 9 and the sum of their squares is 35.

Find three numbers in an A.P. whose sum is 21 and the product of the last two numbers is 63.

Find three numbers in an A.P. whose sum is 12, and product is 60.

Find three numbers in A.P. whose sum is 9 and sum of whose cubes in 99.

The internal angles of a triangle are in A.P. If the smallest angle is 45°, find the remaining angles.

Find 4 numbers in A.P. whose sum is 4 and sum of whose squares is 84.

Find the sum of 50 terms of the A.P. 1 + 4 + 7 + ..... .

Find the sum of 25 terms of the A.P. 8 + 5 + 2 + .... .

Find the sum of the first 200 even natural numbers.

Find the sum of all numbers lying between 201 and 424 which are divisible by 5.

Find the sum of all numbers from 1 to 200 which are divisible by either 2 or 3.

Find the sum of all odd numbers lying between 101 and 200 which are divisible by 3.

Find the sum of all even numbers between 50 and 100 using a formula.

Find the value of x if 1 + 6 + 11 + ... + x = 189.

Find the value of x if 3 + 6 + 9 + ... + 96 = x.

How many terms of the A.P. 6 + 10 + 14 + ... has the sum 880?

How many terms of the A.P. 3 + 9 + 15 + ... has the sum 7500?

The sum of ‘n’ terms of a progression is n(n + 1). Prove that it is an A.P. Also, find its 10th term.

The sum of ‘n’ terms of a progression is (3n² − 5n). Prove that it is an A.Р.

If the sum of ‘n’ terms of a series is (5n² + 3n), then find its first five terms.

The sum of 5 and 15 terms of an A.P. are equal. Find the sum of 20 terms of this A.P.

The sum of 20 and 28 terms of an A.P. are equal. Find the sum of 48 terms of this A.P.

The 4th term of an A.P. is 22, and the 15th term is 66. Find the first term and the common difference. Hence, find the sum of the series to 8 terms.

The sum of 15 terms of an A.P. is zero. Its 4th term is 12. Find its 14th term.

The common difference, last term, and sum of terms of an A.P. are 4, 31, and 136, respectively. Find the number of terms.

In an A.P., the 4th and 6th terms are 6 and 14, respectively. Find the sum of its 20 terms.

The 6th term of an A.P. is equal to 4 times its first term, and the sum of the first 6 terms is 75. Find the first term and the common difference.

In an A.P., if T₁ + T₅ + T₁₀ + T₁₅ + T₂₀ + T₂₄ = 225, find the sum of its 24 terms.

The nth term of an A.P. is (5n − 1). Find the sum of its ‘n’ terms.

The sum of 8 terms of an A.P. is 64, and the sum of 17 terms is 289. Find the sum of its ‘n’ terms.

In an A.P., T₁₂ = 37, d = 3, find a and S₁₂.

The sum of 15 terms of an A.P. is zero, and its 5th term is 12. Find its 12th term.

The nth term of an Arithmetic Progression (A.P.) is given by the relation T_n = 6(7 − n). Find:

1. its first term and the common difference
2. sum of its first 25 terms

The n^th term of a progression is `3^(n + 1)`. Show that it is a G.P. Also, find its 5th term.

Find the 7th term of the G.P. 4, 8, 16, ......

Find the 9th term of the G.P. 2, 1, `1/2`, ......

Find the 8 th term of the G.p. `sqrt3, 1/sqrt3, 1/(3sqrt3)`, .... 

Find the number of terms in the G.P. 1, 2, 4, 8, .... 4096.

Find the number of terms in the G.P. 1, −3, 9, .... −2187.

Find the 5th term from the end of the G.P. `1/512, 1/256, 1/128`, ...256.

Find the 4th term from the end of the G.P. `5/2, 15/8, 45/32, .... 10935/32768`

Which term of the G.P.: `sqrt3, 3, 3sqrt3`, ... is 729?

Which term of the progression 2, 8, 32, ... is 131072?

If the nth terms of the progression 5, 10, 20, ... and the progression 1280, 640, 320, ... are equal, then find the value of n.

The 3rd, 7th, and 11th terms of a G.P. are x, y, and z, respectively, then prove that y² = xz.

The 3rd and 6th terms of a G.P. are 40 and 320, then find the progression.

Find the G.P. whose 2nd and 5th terms are `-3/2 "and" 81/16` respectively.

The (p + q)th and (p − q)th terms of a G.P. are m and n, respectively. Prove that its pth term is `sqrt(mn)` and gth term is `m(n/m)^(p/(2q))`.

Find the G.P. whose 2nd term is 12 and 6th term is 27 times the 3rd term.

The first term of a G.P. is –3. If the 4th term of this G.P. is the square of the 2nd term, then find its 7th term.

The fourth term, the seventh term and the last term of a geometric progression are 10, 80 and 2560 respectively. Find its first term, common ratio and number of terms.

Find the 4 terms in G.P. in which the 3rd term is 9 more than the first term, and the 2nd term is 18 more than the 4th term.

If k, k + 1, and k + 3 are in G.P., then find the value of k.

The product of 3^rd and 8^th terms of a G.P. is 243. If its 4^th term is 3, find its 7^th term.

Find three numbers in G.P. whose sum is 19 and product is 216.

Find three consecutive numbers in G.P. whose sum is 28 and product is 512.

The sum of 3 numbers in a G.P. is 19, and the sum of their squares is 133. Find the numbers.

The product of three consecutive numbers in G.P. is 27 and the sum of the products of numbers taken in pair is 39. Find the numbers.

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

Four numbers are in G.P. The sum of the first two numbers is 4, and the sum of the last two numbers is 36. Find the numbers.

Find the sum of 6 terms of the series 2 + 6 + 18 + .....

Find the sum of 7 terms of the series `16/27 - 8/9 + 4/3 -` ....

Find the sum of 10 terms of the series `1 + sqrt3 + 3` + .....

Find the sum of 7 terms of the series 2 + 0.2 + 0.02 + ....

How many terms of the series 1 + 2 + 4 + .... has the sum 511?

How many terms of the series `2/3 - 1 + 3/2` .... has the sum `463/96`?

The n^th term of a G.P. is 3.(–2)ⁿ. Find the sum of its 7 terms.

The common ratio, last term and sum of n terms of a G.P. are 2, 128 and 255, respectively. Find the value of n.

The ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.

The sum of the first three terms of a G.P. is `1/8` of the sum of the next three terms. Find the common ratio of G.P.

The first and last terms of a Geometrical Progression (G.P.) are 3 and 96, respectively. If the common ratio is 2, find:

1. ‘n’ the number of terms of the G.P.
2. Sum of the n terms.

In a G.P., a = 2, T_n = 162 and S_n = 242. Find the value of n.

15, 30, 60, 120.... are in G.P. (Geometric Progression):

1. Find the n^th term of this G.P. in terms of n.
2. How many terms of the above G.P. will give the sum 945?

The n^th term of an Arithmetic Progression (A.P.) is 2n + 5. The 10^th term is ______.

The sum of 25 terms of A.P. 8 + 12 + 16 + .... is ______.

The 7th term from the end of A.P. 3 + 8 + 13 + .... + 63 is ______.

Which term of A.P. 7 + 10 + 13 + .... is 109?

If k + 6, 2k + 6 and 5k − 2 are in A.P., then the value of k is ______.

If k − 1, k + 1 and k + 5 are in G.P. then the value of k is ______.

The next term of G.P. 4 + 12 + 36 + .... is ______.

The middle most term of the A.P. 3, 8, 13, ....., 63 is ______.

The 5th term from the end of G.P. 2, 4, 8, ..... 4096 is ______.

The 4th term of a G.P. is 16, and the 7th term is 128. Its first term is ______.

The 7^th term of the given Arithmetic Progression (A.P.) `1/a, (1/a + 1), (1/a + 2)`... is ______.

Assertion: 2 + 4 + 6 + 8 + ... + 50 = 650

Reason: Sum of n terms of A.P. `n/2[2a + (n - 1)d]`.

Assertion: The 7th term of the progression `1/4, 1/2`, 1, ... is 32.

Reason: n^th term of G.P. = ar^{n − 1}.

Assertion: If T_n = 3n + 7 for a progression, then T₅ = 22.

Reason: The n^th term of AP = a + (n − 1)d.

Assertion: The sum of 5 terms of G.P. `2/9 - 1/3 + 1/2` ..... is `55/72`.

Reason: The sum of n terms of GP = `n/2(a + r)`.

Assertion: If S_n denotes the sum of n terms of an A.P., then S₁₂ = (S₈ − S₄).

Reason: For an A. P., `S_n = n/2[2a + (n - 1)d]`

In the following questions, two statements (i) and (ii) are given. Choose the valid statement.

1. In an A.P., T₁₂ = 37, d = 3 then a = 4.
2. If 1 + 6 + 11 + ... + x = 189 then x = 41.

In the following questions, two statements (i) and (ii) are given. Choose the valid statement.

1. 7 + 10.5 + 14 + ... + 91 = 1125
2. The number of terms in the progression 8 + 12 + 16 + ... + 124 is 20.

In the following questions, two statements (i) and (ii) are given. Choose the valid statement.

1. The sum of three numbers in A.P. is 90. The middle term will be 30.
2. n^th term from the end in an A.P. = l + (n – 1)d.

In the following questions, two statements (i) and (ii) are given. Choose the valid statement.

1. The 6^th term of the progression 2, 6, 18, ... is 162.
2. Sum of n terms of GP = `(a(1 - r^n))/(1 - r)`.

How many numbers of two digits are divisible by 5?

The 5th term of an A.P. is thrice the second term, and the 12th term exceeds twice the 6th term by 1. Find the 16th term.

If the sum of the first n terms of an A.P. is given by S_n = 3n² + 2n, find the nth term of the A.P.

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

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 ratio of the sum of the first three terms to that of the first 6 terms of a G.P. is 125 : 152. Find the common ratio.

How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?

Determine the number of terms of a G.P. if the first term = 3, the n^th term = 96 and the sum of n terms = 189.