Solutions of chapter 20. Geometric Progressions | RD Sharma - Mathematics

1

Show that each one of the following progressions is a G.P. Also, find the common ratio in each case :

i. 4, -2, 1, ,….

ii.

iii.

iv.

2

Show that the sequence defined by n ∈ N is a G.P.

3

Find:

the ninth term of the G.P. 1, 4, 16, 64, ….

3

Find:

the 10^th term of the G.P.

3

Find:

the 8^th term of the G.P., 0.3, 0.06, 0.012, ….

3

Find:

the 12th term of the G.P. ax, a⁵, x⁵, ….

3

Find:

nth term of the G.P.

3

Find:

the 10^th term of the G.P.

4

Find the 4^th term from the end of the G.P.

5

which term of the progression 0.004, 0.02, 0.1, …. Is 12.5?

6

Which term of the G.P. :

… is ?

6

Which term of the G.P. :

2, 2 √2,4, …. Is 128?

6

Which term of the G.P. :

is 729?

6

Which term of the G.P. :

… is

7

Which term of the progression 18, -12, 8, … is

8

Find the 4th term from the end of the G.P.

9

The fourth term of a G.P. is 27, and the 7th term is 729, find the G.P.

10

The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.

11

If the G.P.’s 5, 10, 20, …. And 1280, 640, 320, … have their nth terms equal, find the value of n.

12

If 5^th, 8^th and 11^th terms of a G.P. are p, q and s respectively, prove that a² = ps.

13

The 4th term of a G.P. is square of its second term, and the first term is -3. Find its 7th term.

14

In a GP the 3^rd term is 24, and the 6^th term is 192. Find the 10^th term.

15

If a, b, c, d and p are different real numbers such that :

(a² + b² + c²)p² – 2(ab + bc + cd) p + (b² + c² + d²) ≤ 0, then show that a, b, c and d are in G.P.

16

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

17

If the p^th and q^th terms of a G.P. are q and p respectively, show that (p + q)^th term is

1

Find three numbers in G.P. whose sum is 65 and whose product is 3375.

2

Find three number in G.P. whose sum is 38 and their product is 1728.

3

The sum of first three terms of a G.P. is , and their product is – 1. Find the G.P.

4

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is . Find them.

5

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

6

The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.

7

The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.

8

Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.

9

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

1

Find the sum of the following geometric progressions :

2, 6, 18, … to 7 terms

1

Find the sum of the following geometric progressions :

1, 3, 9, 27, … to 8 terms

1

Find the sum of the following geometric progressions :

1

Find the sum of the following geometric progressions :

to n terms

1

Find the sum of the following geometric progressions :

4, 2, 1, ….. to 10 terms.

2

Find the sum of the following geometric series :

0.15 + 0.015 + 0.0015 + … to 8 terms;

2

Find the sum of the following geometric series :

to 8 terms ;

2

Find the sum of the following geometric series :

to 5 terms ;

2

Find the sum of the following geometric series :

(x + y) + (x² + xy + y²) + (x³ + x² y + xy² + y³) + …. to n terms ;

2

Find the sum of the following geometric series :

to 2n terms;

2

Find the sum of the following geometric series :

2

Find the sum of the following geometric series :

1, - a, a², - a³, …. to n terms (a ≠1)

2

Find the sum of the following geometric series :

x³, x⁵, x⁷, … to n terms

2

Find the sum of the following geometric series :

to n terms

3

Evaluate the following :

3

Evaluate the following :

3

Evaluate the following :

4

Find the sum of the following series :

5 + 55 + 555 + … to n terms.

4

Find the sum of the following series :

7 + 77 + 777 + … to n terms.

4

Find the sum of the following series :

9 + 99 + 999 + … to n terms.

4

Find the sum of the following series :

0.5 + 0.55 + 0.555 + …. to n terms

4

Find the sum of the following series :

0.6 + 0.66 + 0.666 + …. to n terms.

5

How many terms of the G.P. . Be taken together to make

6

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

7

How many terms of the sequence must be taken to make the sum ?

8

The sum of n terms of the G.P. 3, 6, 12, … is 381. Find the value of n.

9

The common ratio of a G.P. is 3, and the last term is 486. If the sum of these terms be 728, find the first term.

10

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

11

The 4th and 7th terms of a G.P. are and respectively. Find the sum of n terms of the G.P.

12

Find the sum :

13

The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.

14

If S₁, S₂, S₃ be respectively the sums of n, 2n, 3n terms of a G.P., then prove that S₁² + S₂² = S₁(S₂ + S₃)

_{Question May be wrong.}

15

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is

16

If a and b are the roots of x² – 3x + p = 0 and c, d are the roots x² – 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17 : 15.

17

How many terms of the G.P. are needed to give the sum ?

18

A person has 2 parents, 4 grandparents, 8 great grand parents, and so on. Find the number of his ancestors during the ten generations preceding his own.

19

If S₁, S₂, …., S_n are the sums of n terms of n G.P.’s whose first term is 1 in each and common ratios are 1, 2, 3, …., n respectively, then prove that

S₁ + S₂ + 2S₃ + 3S₄ + … (n – 1) S_n = 1ⁿ + 2ⁿ + 3ⁿ + … + nⁿ.

20

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.

21

Let a_n be the nth term of the G.P. of positive numbers. Let and such that α ≠ β. Prove that the common ratio of the G.P. is α /β

22

Find the sum of 2n terms of the series whose every even term is ‘a’ times the term before it and every odd term is ‘c’ times the term before it, the first term being unity.

1

Find the sum of the following series to infinity :

1

Find the sum of the following series to infinity :

8 + + 4 + …. ∞

1

Find the sum of the following series to infinity :

2/5 + 3/5² + 2/5³ + 3/5⁴ + …. ∞

1

Find the sum of the following series to infinity :

10 – 9 + 8.1 – 7.29 + …. ∞

1

Find the sum of the following series to infinity :

2

Prove that :

(9^1/3 . 9^1/9 . 9^1/27 ….∞) = 3.

3

Prove that :

(2^1/4 .4^1/8 . 8^1/16. 16^1/32….∞) = 2.

4

If S_p denotes the sum of the series 1 + r^p + r^2p + … to ∞ and s_p the sum of the series 1 – r^p + r^2p - … to ∞, prove that s_p + S_p = 2 S_2p.

5

Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.

6

Express the recurring decimal 0.125125125 … as a rational number.

7

Find the rational number whose decimal expansion is

8

Find the rational numbers having the following decimal expansions :

8

Find the rational numbers having the following decimal expansions :

8

Find the rational numbers having the following decimal expansions :

8

Find the rational numbers having the following decimal expansions :

9

One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.

10

Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.

11

The sum of the first two terms of an infinite G.P. is 5, and each term is three times the sum of the succeeding terms. Find the G.P.

12

Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.

13

If S denotes the sum of an infinite G.P. and S₁ denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively and

1

If a, b, c are in G.P., prove that log a, log b, log c are in A.P.

2

If a, b, c are in G.P., prove that are in A.P.

3

Find k such that k + 9, k – 6 and 4 form three consecutive terms of a G.P.

4

Three numbers are in A.P., and their sum is 15. If 1, 3, 9 be added to them respectively, they from a G.P. Find the numbers.

5

The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

6

The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.

7

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 A.P. Find the numbers.

8

If a, b, c are in G.P., prove that :

a(b² + c²) = c(a² + b²)

8

If a, b, c are in G.P., prove that :

8

If a, b, c are in G.P., prove that :

8

If a, b, c are in G.P., prove that :

8

If a, b, c are in G.P., prove that :

(a + 2b + 2c) (a – 2b + 2c) = a² + 4c².

9

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

9

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

(a + b + c + d)² = (a + b)² + 2(b + c)² + (c + d)²

9

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

(b + c) (b + d) = (c + a) (c + d)

10

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

a², b², c²

10

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

a³, b³, c³

10

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

a² + b², ab + bc, b² + c²

11

If a, b, c are in G.P., prove that :

(a² + b²), (b² + c²), (c² + d²) are in G.P.

11

If a, b, c are in G.P., prove that :

(a² – b²), (b² – c²), (c² – d²) are in G.P.

11

If a, b, c are in G.P., prove that :

are in G.P

11

If a, b, c are in G.P., prove that :

(a² + b² + c²), (ab + bc + cd), (b² + c² + d²) are in G.P.

12

If (a – b), (b – c), (c – a) are in G.P., then prove that (a + b + c)² = 3(ab + bc + ca)

13

If a, b, c are in G.P., then prove that :

14

If the 4^th, 10^th and 16^thterms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.

15

If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, a – b, d – c are in G.P.

16

If pth, qth, rth and sth terms of an A.P., be in G.P., then prove that p – q, q – r, r – s are in G.P.

17

If are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.

18

If x^a = x^b/2z^b/2 = z^c, then prove that are in A.P.

19

If a, b, c are in A.P. b, c, d are in G.P. and are in A.P., prove that a, c, e are in G.P.

20

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x², b², y² are in A.P.

21

If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a – b), (d – c) are in G.P.

22

If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x < – 1 or x > 3.

23

If p^th, q^th and r^th terms of an A.P. and G.P. are both a, b and c respectively, show that a^{b – c}b^c ^{– a} c^{a – b} = 1.

1

Insert 6 geometric means between 27 and .

2

Insert 5 geometric means between 16 and .

3

Insert 5 geometric means between and .

4

Find the geometric means of the following pairs of numbers :

i. 2 and 8

ii. a³b and ab³

iii. –8 and –2

5

If a is the G.M. of 2 and find a.

6

Find the two numbers whose A.M. is 25 and GM is 20.

7

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

8

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio (3 + 2) : (3 – 2).

9

If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.

10

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

11

Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.

12

If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that : a : b = (2 + ) : (2 – ).

13

If one A.M., A and two geometric means G₁ and G₂ inserted between any two positive numbers, show that

1

If the fifth term of a G.P. is 2, then write the product of its 9 terms.

2

If and terms of a G.P. are m and n respectively, then write its pth term.

3

If and x are in G.P., then write the value of x.

4

If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is , then write its first term and common difference.

5

If and terms of a G.P. are x, y, z respectively, then write the value of .

6

If A₁, A₂ be two AM’s and G₁, G₂ be two GM’s between a and b, then find the value of .

7

If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.

8

Write the quadratic equation the arithmetic and geometric means of whose roots are A and G respectively.

9

Write the product of n geometric means between two number a and b

10

If , then write b in terms of a given that .