## Selina Concise Mathematics Class 10 ICSE Solutions Geometric Progression

**Selina Publishers Concise Mathematics Class 10 ICSE Solutions Chapter 11 Geometric Progression**

### Geometric Progression Exercise 11A – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.

Find, which of the following sequence form a G.P. :

(i) 8, 24, 72, 216, ……

(ii) \(\frac{1}{8}, \frac{1}{24}, \frac{1}{72}, \frac{1}{216}\), ……..

(iii) 9, 12, 16, 24, ……

Solution 1(i).

Solution 1(ii).

Solution 1(iii).

Question 2.

Find the 9th term of the series :

1, 4, 16, 64 ……..

Solution:

Question 3.

Find the seventh term of the G.P. :

1, \(\sqrt{3}\), 3, \(3 \sqrt{3}\) …..

Solution:

Question 4.

Find the 8^{th} term of the sequence :

\(\frac{3}{4}, 1 \frac{1}{2}\) 3, …….

Solution:

Question 5.

Find the 10^{th} term of the G.P. :

Solution:

Question 6.

Find the n^{th} term of the series :

Solution:

Question 7.

Find the next three terms of the sequence :

\(\sqrt{5}\), 5, \(5 \sqrt{5}\), ……

Solution:

Question 8.

Find the sixth term of the series :

2^{2}, 2^{3}, 2^{4}, ……….

Solution:

Question 9.

Find the seventh term of the G.P. :

[late]\sqrt{3}+1,1, \frac{\sqrt{3}-1}{2}[/latex], ……………..

Solution:

Question 10.

Find the G.P. whose first term is 64 and next term is 32.

Solution:

Question 11.

Find the next three terms of the series:

\(\frac{2}{27}, \frac{2}{9}, \frac{2}{3}\), ………….

Solution:

Question 12.

Find the next two terms of the series

2 – 6 + 18 – 54 …………

Solution:

### Geometric Progression Exercise 11B – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.

Which term of the G.P. :

Solution:

Question 2.

The fifth term of a G.P. is 81 and its second term is 24. Find the geometric progression.

Solution:

Question 3.

Fourth and seventh terms of a G.P. are \(\frac{1}{18} \text { and }-\frac{1}{486}\) respectively. Find the GP.

Solution:

Question 4.

If the first and the third terms of a G.P. are 2 and 8 respectively, find its second term.

Solution:

Question 5.

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

Solution:

Question 6.

Find the geometric progression with 4^{th} term = 54 and 7^{th} term = 1458.

Solution:

Question 7.

Second term of a geometric progression is 6 and its fifth term is 9 times of its third term. Find the geometric progression. Consider that each term of the G.P. is positive.

Solution:

Question 8.

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.

Solution:

Question 9.

If the 4th and 9th terms of a G.P. are 54 and 13122 respectively, find the GP. Also, find its general term.

Solution:

Question 10.

The fifth, eight and eleventh terms of a geometric progression are p, q and r respectively. Show that : q^{2} = pr.

Solution:

### Geometric Progression Exercise 11C – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.

Find the seventh term from the end of the series : \(\sqrt{2}\) , 2, \(2 \sqrt{2}\), ………. 32.

Solution:

Question 2.

Find the third term from the end of the GP.

\(\frac{2}{27}, \frac{2}{9}, \frac{2}{3}\), ………….. 162

Solution:

Question 3.

For the \(\frac{1}{27}, \frac{1}{9}, \frac{1}{3}\), ………… 81;

find the product of fourth term from the beginning and the fourth term from the end.

Solution:

Question 4.

If for a G.P., p^{th}, q^{th} and r^{th} terms are a, b and c respectively ; prove that :

(q – r) log a + (r – p) log b + (p – q) log c = 0

Solution:

Question 5.

If a, b and c in G.P., prove that : log a^{n}, log b^{n} and log c^{n} are in A.P.

Solution:

Question 6.

If each term of a G.P. is raised to the power x, show that the resulting sequence is also a G.P.

Solution:

Question 7.

If a, b and c are in A.P. a, x, b are in G.P. whereas b, y and c are also in G.P. Show that : x^{2}, b^{2}, y^{2} are in A.P.

Solution:

Question 8.

If a, b, c are in G.P. and a, x, b, y, c are in A.P., prove that :

Solution 8(i).

Solution 8(ii).

Question 9.

If a, b and c are in A.P. and also in G.P., show that: a = b = c.

Solution:

Question 10.

The first term of a G.P. is a and its n^{th} term is b, where n is an even number.If the product of first n numbers of this G.P. is P ; prove that : p^{2} – (ab)^{n}.

Solution:

Question 11.

If a, b, c and d are consecutive terms of a G.P. ; prove that :

(a^{2} + b^{2}), (b^{2} + c^{2}) and (c^{2} + d^{2}) are in GP.

Solution:

Question 12.

If a, b, c and d are consecutive terms of a G.P. To prove:

Solution:

### Geometric Progression Exercise 11D – Selina Concise Mathematics Class 10 ICSE Solutions

Question 1.

Find the sum of G.P. :

(i) 1 + 3 + 9 + 27 + ……….. to 12 terms.

(ii) 0.3 + 0.03 + 0.003 + 0.0003 + …… to 8 terms.

Solution 1(i).

Solution 1(ii).

Solution 1(iii).

Solution 1(iv).

Solution 1(v).

Solution 1(vi).

Question 2.

How many terms of the geometric progression 1+4 + 16 + 64 + ……… must be added to get sum equal to 5461?

Solution:

Question 3.

The first term of a G.P. is 27 and its 8^{th} term is \(\frac{1}{81}\). Find the sum of its first 10 terms.

Solution:

Question 4.

A boy spends ₹ 10 on first day, ₹ 20 on second day, ₹ 40 on third day and so on. Find how much, in all, will he spend in 12 days?

Solution:

Question 5.

The 4th and the 7th terms of a G.P. are \(\frac{1}{27} \text { and } \frac{1}{729}\) respectively. Find the sum of n terms of this G.P.

Solution:

Question 6.

A geometric progression has common ratio = 3 and last term = 486. If the sum of its terms is 728 ; find its first term.

Solution:

Question 7.

Find the sum of G.P. : 3, 6, 12, ……………. 1536.

Solution:

Question 8.

How many terms of the series 2 + 6 + 18 + ………….. must be taken to make the sum equal to 728 ?

Solution:

Question 9.

In a G.P., the ratio between the sum of first three terms and that of the first six terms is 125 : 152.

Find its common ratio.

Solution:

Question 10.

Find how many terms of G.P. \(\frac{2}{9}-\frac{1}{3}+\frac{1}{2}\) ………. must be added to get the sum equal to \(\frac{55}{72}\)?

Solution:

Question 11.

If the sum 1 + 2 + 2^{2} + ………. + 2^{n-1} is 255, find the value of n.

Solution:

Question 12.

Find the geometric mean between :

(i) \(\frac{4}{9} \text { and } \frac{9}{4}\)

(ii) 14 and \(\frac{7}{32}\)

(iii) 2a and 8a^{3}

Solution 12(i).

Solution 12(ii).

Solution 12(iii).

Question 13.

The sum of three numbers in G.P. is \(\frac{39}{10}\) and their product is 1. Find the numbers.

Solution:

Question 14.

The first term of a G.P. is -3 and the square of the second term is equal to its 4^{th} term. Find its 7^{th} term.

Solution:

Question 15.

Find the 5^{th} term of the G.P. \(\frac{5}{2}\), 1, …..

Solution:

Question 16.

The first two terms of a G.P. are 125 and 25 respectively. Find the 5th and the 6th terms of the G.P.

Solution:

Question 17.

Find the sum of the sequence –\(\frac{1}{3}\), 1, – 3, 9, …………. upto 8 terms.

Solution:

Question 18.

The first term of a G.P. in 27. If the 8thterm be \(\frac{1}{81}\), what will be the sum of 10 terms ?

Solution:

Question 19.

Find a G.P. for which the sum of first two terms is -4 and the fifth term is 4 times the third term.

Solution:

**Additional Questions**

Question 1.

Find the sum of n terms of the series :

(i) 4 + 44 + 444 + ………

(ii) 0.8 + 0.88 + 0.888 + …………..

Solution 1(i).

Solution 1(ii).

Question 2.

Find the sum of infinite terms of each of the following geometric progression:

Solution 2(i).

Solution 2(ii).

Solution 2(iii).

Solution 2(iv).

Solution 2(v).

Question 3.

The second term of a G.P. is 9 and sum of its infinite terms is 48. Find its first three terms.

Solution:

Question 4.

Find three geometric means between \(\frac{1}{3}\) and 432.

Solution:

Question 5.

Find :

(i) two geometric means between 2 and 16

(ii) four geometric means between 3 and 96.

(iii) five geometric means between \(3 \frac{5}{9}\) and \(40 \frac{1}{2}\)

Solution 5(i).

Solution 5(ii).

Solution 5(iii).

Question 6.

The sum of three numbers in G.P. is \(\frac{39}{10}\) and their product is 1. Find the numbers.

Solution:

Sum of three numbers in G.P. = \(\frac{39}{10}\) and their product = 1

Let number be \(\frac{a}{r}\), a, ar, then

Question 7.

Find the numbers in G.P. whose sum is 52 and the sum of whose product in pairs is 624.

Solution:

Question 8.

The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.

Solution:

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