ICSE Solutions for Selina Concise Chapter 11 Geometric Progressions Class 10 Maths
Exercise 11(A)
1. Find, which of the following sequence form a G.P. :
(i) 8, 24, 72, 216, ….
(ii) 1/8,1/24,1/72,1/216, ……
(iii) 9, 12, 16, 24, …
Solution
(i)
(ii)
(iii)
2. Find the 9th term of the series :
1, 4, 16, 64 …
Solution
3. Find the seventh term of the G.P. :
1, √3, 3, 3√3…..
Solution
4. Find the 8th term of the sequence :
34,112 3, ……
Solution
5. Find the 10th term of the G.P. :
Solution
6. Find the nth term of the series :
Solution
7. Find the next three terms of the sequence :
√5, 5, 5√5, ……
Solution
8. Find the sixth term of the series :
22, 23, 24, ……
Solution
9. Find the seventh term of the G.P. :
√3+1, 1, (√3−1)/2, ……
Solution
10. Find the G.P. whose first term is 64 and next term is 32.
Solution
11. Find the next three terms of the series:
2/27,2/9,2/3, ……
Solution
12. Find the next two terms of the series
2 – 6 + 18 – 54 ……
Solution
Exercise 11(B)
1. Which term of the G.P. : − 10, 5/√3, −5/6,……. is −5/72 ?
Solution
2. The fifth term of a G.P. is 81 and its second term is 24. Find the geometric progression.
Solution
3. Fourth and seventh terms of a G.P. are 1/18 and −1/486 respectively. Find the GP.
Solution
4. If the first and the third terms of a G.P. are 2 and 8 respectively, find its second term.
Solution
5. The product of 3rd and 8th terms of a G.P. is 243. If its 4th term is 3, find its 7th term.
Solution
6. Find the geometric progression with 4th term = 54 and 7th term = 1458.
Solution
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
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
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
10. The fifth, eight and eleventh terms of a geometric progression are p, q and r respectively. Show that : q2 = pr.
Solution
Exercise 11(C)
1. Find the seventh term from the end of the series : √2, 2, 2√2, …. 32.
Solution
2. Find the third term from the end of the GP.
2/27, 2/9, 2/3, …… 162
Solution
3. For the 1/27,1/9,1/3, …… 81;
find the product of fourth term from the beginning and the fourth term from the end.
Solution
4. If for a G.P., pth, qth and rth terms are a, b and c respectively ; prove that :
(q – r) log a + (r – p) log b + (p – q) log c = 0
Solution:
5. If a, b and c in G.P., prove that : log an, log bn and log cn are in A.P.
Solution
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
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 : x2, b2, y2 are in A.P.
Solution
8. If a, b, c are in G.P. and a, x, b, y, c are in A.P., prove that :
(i) 1/x + 1/y = 2/b
(ii) a/x + c/y = 2
Solution
(i)
(ii)
9. If a, b and c are in A.P. and also in G.P., show that: a = b = c.
Solution
10. The first term of a G.P. is a and its nth term is b, where n is an even number.If the product of first n numbers of this G.P. is P ; prove that : p2 – (ab)n.
Solution
11. If a, b, c and d are consecutive terms of a G.P. ; prove that :
(a2 + b2), (b2 + c2) and (c2 + d2) are in GP.
Solution
12. If a, b, c and d are consecutive terms of a G.P. To prove:
1/(a2 + b2), 1/(b2 + c2) and 1/(c2 + d2) are in G.P.
Solution
Exercise 11(D)
(i) 1 + 3 + 9 + 27 + ……. to 12 terms.
(ii) 0.3 + 0.03 + 0.003 + 0.0003 + …… to 8 terms.
(iii) 1 −1/2 + 1/4 −1/8 + ……. to 9 terms.
(iv) 1 −1/3 + 1/32 −1/32 + ……. to n terms.
(v) (x+y)/(x-y) + 1 + (x-y)/(x+y) + ……. upto n terms.
(vi) √3 +1/√3 + 1/3√3 + ……. to n terms.
Solution
(i)
(ii)
(iii)
(iv)
(v)
(vi).
2. How many terms of the geometric progression 1+4 + 16 + 64 + ……… must be added to get sum equal to 5461?
Solution
3. The first term of a G.P. is 27 and its 8th term is 1/81. Find the sum of its first 10 terms.
Solution
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
5. The 4th and the 7th terms of a G.P. are 1/27 and 1/729 respectively. Find the sum of n terms of this G.P.
Solution
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
7. Find the sum of G.P. : 3, 6, 12, …... 1536.
Solution
8. How many terms of the series 2 + 6 + 18 + …... must be taken to make the sum equal to 728 ?
Solution
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
10. Find how many terms of G.P. 2/9−1/3+1/2 ………. must be added to get the sum equal to 55/72?
Solution
11. If the sum 1 + 2 + 22 + ……. + 2n-1 is 255, find the value of n.
Solution
12. Find the geometric mean between :
(i) 4/9 and 9/4
(ii) 14 and 7/32
(iii) 2a and 8a3
Solution
(i)
(ii)
(iii)
13. The sum of three numbers in G.P. is 39/10 and their product is 1. Find the numbers.
Solution
14. The first term of a G.P. is -3 and the square of the second term is equal to its 4th term. Find its 7th term.
Solution
15. Find the 5th term of the G.P. 5/2, 1, …..
Solution
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
17. Find the sum of the sequence –1/3, 1, – 3, 9, …… upto 8 terms.
Solution
18. The first term of a G.P. in 27. If the 8thterm be 1/81, what will be the sum of 10 terms ?
Solution
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
1. Find the sum of n terms of the series :
(i) 4 + 44 + 444 + ……
(ii) 0.8 + 0.88 + 0.888 + ……
Solution
(i)
(ii)
2. Find the sum of infinite terms of each of the following geometric progression:
(i) 1+ 1/3 + 1/9 + 1/27 + ……
(ii) 1 – 1/2 + 1/4 – 1/8 + …….
(iii) 1/3 + 1/32 – 1/33 + ……
(iv) √2 – 1/√2 + 1/2√2 – 1/4√2 + ……
(v) √3 + 1/√3 + 1/3√3 + 1/9√3 + ……..
Solution
(i)
(ii)
(iii)
(iv)
(v)
3. The second term of a G.P. is 9 and sum of its infinite terms is 48. Find its first three terms.
Solution
4. Find three geometric means between 13 and 432.
Solution
5. Find :
(i) two geometric means between 2 and 16
(ii) four geometric means between 3 and 96.
(iii) five geometric means between 3(5/9) and 40(1/2)
Solution
(i)
(ii)
(iii).
6. The sum of three numbers in G.P. is 39/10 and their product is 1. Find the numbers.
Solution
Sum of three numbers in G.P. = 39/10 and their product = 1
Let number be a/r, a, ar, then
7. Find the numbers in G.P. whose sum is 52 and the sum of whose product in pairs is 624.
Solution
8. The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.
Solution