M l aggarwal understanding icse mathematics class 10 Solutions for Chapter 8 arithmetic and geometric progression

For the following A.P.s, write the first term ‘a’ and the common difference ‘d’:

(i) 3, 1, – 1, – 3, …

(ii) 1/3, 5/3, 9/3, 13/3, ….

(iii) -3.2, -3, -2.8, -2.6, ….

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

(i) a = 10, d = 10

(ii) a = -2, d = 0

(iii) a = 4, d = -3

(iv) a = ½, d = -1/6

Which of 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:

(i) 4, 10, 16, 22, …

(ii) -2, 2, -2, 2, …

(iii) 2, 4, 8, 16, …

(iv) 2, 5/2, 3, 7/2, …

(v) – 10, -6, -2, 2, …

(vi) 1^{2}, 3^{2}, 5^{2}, 7^{2}, \ldots

(vii) 1, 3, 9, 27, …

(viii) √2, √8, √18, √32, …

(ix) a, 2a, 3a, 4a, …

(x) a, 2a + 1, 3a + 2, 4a + 3, …

Find the A.P. whose n^{\text {th }} term is 7 – 3K. Also, find the 20th term.

Find the indicated terms in each of the following A.P.s:

(i) 1, 6, 11, 16, …; a_{20}

(ii) -4, -7, -10, -13, …, a_{25}, a_{n}

Find the n^{\text {th }} term and the 12^{\text {th }} term of the list of numbers: 5, 2, – 1, – 4, …

Find the 8^{\text {th }} term of the A.P. whose first term is 7 and the common difference is 3.

(i) If the common difference of an A.P. is – 3 and the 18^{\text {th }} term is – 5, then find its first term.

(ii) If the first term of an A.P. is – 18 and its 10 t h term is zero, then find its common difference.

Which term of the A.P.

(i) 3, 8, 13, 18, … is 78?

(ii) 7, 13, 19, … is 205 ?

(iii) 18, 15½, 13, … is – 47 ?

(i) Check whether – 150 is a term of the A.P. 11, 8, 5, 2, …

(ii) Find whether 55 is a term of the A.P. 7, 10, 13, … or not. If yes, find which term is it.

(iii) Is 0 a term of the A.P. 31, 28, 25,…? Justify your answer.

(i) Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253.

(ii) Find the 12th from the end of the A.P. – 2, – 4, – 6, …, – 100.

Find the sum of the two middlemost terms of the A.P.

-4/3, -1, -2/3, …, 4 \frac{1}{3}

Which term of the A.P. 53, 48, 43,… is the first negative term?

Determine the A.P. whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.

Determine the A.P. whose third term is 16 and the 7^{\text {th }} term exceeds the 5^{\text {th }} term by 12.

Find the 20^{\text {th }} term of the A.P. whose 7^{\text {th }} term is 24 less than the 11^{\text {th }} term, first term being 12.

Find the 31^{\text {st }} term of an A.P. whose 11^{\text {th }} term is 38 and 6^{\text {th }} term is 73.

If the seventh term of an A.P. is 1/9 and its ninth term is 1/7, find its 63rd term.

(i) The 15th term of an A.P. is 3 more than twice its 7th term. If the 10th term of the A.P. is 41, find its nth term.

(ii) The sum of 5th and 7th terms of an A.P. is 52 and the 10th term is 46. Find the A.P.

If the 8th term of an A.P. is zero, prove that its 38th term is triple of its 18th term.

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

If the nth terms of the two A.P.s 9, 7, 5, … and 24, 21, 18, … are the same, find the value of n. Also, find that term.

(i) How many two-digit numbers are divisible by 3?

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

(iii) How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?

If the numbers n – 2, 4n – 1 and 5n + 2 are in A.P., find the value of n.

The sum of three numbers in A.P. is 3 and their product is – 35. Find the numbers.

The sum of three numbers in A.P. is 30 and the ratio of the first number to the third number is 3: 7. Find the numbers.

Find the sum of the following A.P.s :

(i) 2, 7, 12, … to 10 terms

(ii) 1/15, 1/12, 1/10, … to 11 terms

Find the sums given below :

(i) 34 + 32 + 30 + … + 10

(ii) – 5 + ( – 8) + ( – 11) + … + ( – 230)

In an A.P. (with usual notations) :

(i) given a = 5, d = 3, a_{n} = 50, find n and \mathrm{S}_{\mathrm{n}}

(ii) given a = 7, a_{13} = 35, find d and \mathrm{S}_{13}

(iii) given d = 5, \mathrm{S}_{9} = 75, find a and \mathbf{a}_{9}

(iv) given a = 8, a_{n}=62, S_{n}=210, find n and d

(v) given a = 3, n = 8, S = 192, find d.

Solve for x : 1 + 4 + 7 + 10 + … + x = 287.

(i) How many terms of the A.P. 25, 22, 19, … are needed to give the sum 116? Also, find the last term.

(ii) How many terms of the A.P. 24, 21, 18, … must be taken so that the sum is 78 ? Explain the double answer.

(i) Find the sum of the first 51 terms of the A.P. whose second and third terms are 14 and 18 respectively.

(ii) If the third term of an A.P. is 1 and 6th term is – 11, find the sum of its first 32 terms.

If the sum of the first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of the first 10 terms.

Show that \mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots form an A.P. where a_{n} is defined as a_{n} = 3 + 4n. Also find the sum of first 15 terms.

(i) \text { If } a_{n}=3-4 n, \text { show that } a_{1}, a_{2}, a_{3}, \ldots \text { form an A.P. Also find } S_{20}

(ii) Find the common difference of an A.P. whose first term is 5 and the sum of the first four terms is half the sum of the next four terms.

(i) Find the next term of the list of numbers 1/6, 1/3, 2/3, …

(ii) Find the next term of the list of numbers 3/16, -3/8, ¾, -3/2,…

(iii) Find the 15th term of the series √3 + 1/√3 + 1/3√3 + …

(iv) Find the n^{\text {th }} term of the list of numbers 1/√2, -2, 4√2, – 16,…

(v) Find the 10th and nth terms of the list of numbers 5, 25, 125, …

(vi) Find the 6th and the nth terms of the list of numbers 3/2, ¾, 3/8,…

(vii) Find the 6th term from the end of the list of numbers 3, – 6, 12, – 24, …, 12288.

Which term of the G.P. 3, – 3√3, 9, -9√3, … is 729?

Determine the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2.

Find the value of x such that,

(i) -2/7, x, -7/2 are three consecutive terms of a G.P.

(ii) x + 9, x – 6 and 4 are three consecutive terms of a G.P.

(iii) x, x + 3, x + 9 are the first three terms of a G.P. Find the value of x.

If the fourth, seventh, and tenth terms of a G.P. are x, y, z respectively, prove that x, y, z are in G.P.

If a, b, c are in A.P., then show that 3^{a}, 3^{b}, 3^{c} are in G.P.

The first and the second terms of a GP. are x^{-4} \text {and } x^{m}. If its 8th term is x^{52}, then find the value of m.

Find the geometric progression whose 4th term is 54 and the 7th term is 1458.

The sum of the first three terms of a G.P. is 39/10 and their product is 1. Find the common ratio and the terms.

Show that the products of the corresponding terms of two G.P. a, ar, \mathrm{ar}^{2}, \ldots, \operatorname{ar}^{n-1} \text { and } A, A R, A R^{2}, \ldots, A R^{n-1} form a G.P. and find the common ratio.

If a, b, c are in GP., prove that the following are also in G.P.

(i) a^{3}, b^{3}, c^{3}

(ii) a^{2}+b^{2}, a b+b c, b^{2}+c^{2}

If a, b, c, d are in G.P., show that

(i) a^{2}+b^{2}, b^{2}+c^{2}, c^{2}+d^{2} \text { are in G.P. }

(ii) (b-c)^{2}+(c-a)^{2}+(d-b)^{2}=(a-d)^{2}

Find the sum of the first 10 terms of the geometric series √2 + √6 + √18 + ….

The nth term of an A.P. is 6n + 2. Find the common difference.

The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then find its first term.

If the sum of the first six terms of any G.P. is equal to 9 times the sum of the first three terms, then find the common ratio of the G.P.

If the 8th term of an A.P. is 31 and the 15th term is 16 more than its 11th term, then find the A.P.

How many terms of the GP. 1, √2 > 2, 2 √2, … are required to give a sum of 1023( √2 + 1)?

The 19th term of an A.P. is equal to three times its 6th term. If its 9th term is 19, find the A.P.

Which term of the list of numbers 5, 2, – 1, – 4, … is – 55?

Which term of the list of numbers 20, 19¼, 18½, 17¾, … is the first negative term?

Find the third term of a G.P. whose common ratio is 3 and the sum of whose first seven terms is 2186.

How many three-digit numbers are divisible by 9?

If the first term of a G.P. is 5 and the sum of the first three terms is 31/5, find the common ratio.

The sum of three numbers in A.P. is – 3 and the product is 8. Find the numbers.

The angles of a quadrilateral are in A.P. If the greatest angle is double of the smallest angle, find all the four angles.

Find the sum of the series 1+(1+x)+\left(1+x+x^{2}\right)+\ldots to n terms, x ≠ 1.