# Geometric Progression Questions and Answers

## Geometric Progression Questions for Practice

### General form of Geometric Progression:

a, ar, ar², ar³, ……..

where,
The first term is denoted as = a
The common ratio is denoted as = r

### Types of Geometric Progression

Geometric progression can be classified based on the number of terms in the sequence.

• Finite Geometric Progression: A finite geometric progression is a sequence of numbers that has a fixed number of terms. The general form of a finite geometric progression is:
a, ar, ar^2, ar^3, ar^4, …, ar^(n-1)
For example: 2, 4, 8,16
• Infinite Geometric Progression: An infinite geometric progression is a sequence of numbers that goes on forever. The general form of an infinite geometric progression is :
a, ar, ar^2, ar^3, ar^4, …,
For example: 2, 4, 8, …………
Application of Geometric Progression is in physics, engineering, biology, economics, computer science, queueing theory, and finance that’s why Geometric Progression questions and answers are important to know.

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1. How do you find S7 for the geometric series 1 + 9 + 81 + 729 +…?

6,94,765

6,94,765

10.71%

5,97,871

5,97,871

64.29%

2,44,406

2,44,406

5.95%

None of the above

None of the above

19.05%

First term a = 1 and r = 9.

$s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{7} = 1 \times \frac{9^{7} -1}{9-1}$

= 5,97,871

2. Find the sum of the geometric series 3 + 9 + 27 + 81 + . . . where there are 5 terms in the series.

363

363

68.35%

362

362

12.66%

242

242

6.33%

243

243

12.66%

For this series, we have a = 3, r = 3 and n = 5.

$s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{5} = 3 \times \frac{3^{5} -1}{3-1} = 363$

3. Find the sum of the geometric series 8, − 4, + 2, − 1,. . . where there are 7 terms in the series.

5.37

5.37

77.14%

5.50

5.50

8.57%

6.34

6.34

4.29%

None of the above

None of the above

10%

For this series, we have a = 8, r = $\frac{-1}{2}$ and n = 7.

Thus

$s_{n} = a \times \frac{1-r^{n}}{1-r}$ if r<1

$s_{7} = 8 \times \frac{1-(\frac{-1}{2})^{7}}{1-(\frac{-1}{2})}$

S₇ =5.37.

4. How many terms are there in the geometric progression 4, 8, 16, . . ., 512?

5

5

10.29%

6

6

1.47%

7

7

14.71%

8

8

73.53%

Here a = 4 and r = 2. nth term=512. But the formula for the nth term is $ar^{n-1}$

So

512 = $4\times 2^{n-1}$

128= $2^{n-1}$

$2^{7} = 2^{n-1}$

7 = n − 1

n = 8.

5. If the common ratio in a specific series is 3. Last term is 486 and sum of the terms is 728. Then find out the first term of gp?

3

3

5.66%

1

1

15.09%

2

2

71.7%

8

8

7.55%

The series of gp is: 1, ar², ar³,………

Common ratio=3

Last term= 486

So $a_{n} = a\times r^{n-1}$ = 486

$a_{n} = a\times 3^{n-1}$=486

a(3ⁿ) = $486\times 3$ = 1458-----(1)

Now, sum of G.P.=

=  728

By putting the value of r and using eq(1) we get

$\frac{1458-a}{2} = 728$

1458 - a = 1456

So a= 2.

6. How do you find S₁₀ for the geometric series 1 + 5 + 25 + 125 +…?

25,00,000

25,00,000

8.47%

23,87,463

23,87,463

3.39%

24,41,406

24,41,406

81.36%

None of the above

None of the above

6.78%

As we know first term a=1 and r=5.

$s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{10} = 1 \times \frac{5^{10} -1}{5-1}$

= 24,41,406

7. What is the sum of below given infinite g.p.?

$\frac{1}{2} +\frac{1}{3}+\frac{1}{4} +\frac{1}{9} +\frac{1}{8} +\frac{1}{27}+$…………………..=

$\frac{1}{2}$

$\frac{1}{2}$

7.55%

$\frac{2}{3}$

$\frac{2}{3}$

15.09%

$\frac{2}{5}$

$\frac{2}{5}$

7.55%

$\frac{3}{2}$

$\frac{3}{2}$

69.81%

8. How do you find $s_{17}$ for the geometric series 5 + 15 + 45 + 135 + …?

32,28,50,405

32,28,50,405

75.51%

12,91,40,631

12,91,40,631

10.2%

34,2154,214

34,2154,214

6.12%

None of the above

None of the above

8.16%

As we know first term a=5 and r=3.

Formula= $s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{17} = 5 \times \frac{3^{17} -1}{3-1}$

=32,28,50,405

9. How do you find the sum of the first 6 terms of the geometric series: 6+ 36 + 216…?

33454

33454

6.67%

55986

55986

84.44%

43659

43659

6.67%

76463

76463

2.22%

Common ratio = 6 and  First term = 6

As per the formula:

$s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{6} = 6 \times \frac{6^{6} -1}{6-1}$

= 55986

10. Find out the sum of the next infinite geometric series, if it exists? 3 + 2.25 + 1.6875 + 1.265625+…?

10

10

8.51%

11

11

8.51%

12

12

78.72%

13

13

4.26%

Here a=3

r=0.75

Since |r|<1

So = $\frac{a_{1}}{1-r}$

Or, sum= $\frac{3}{(1-0.75)} = 12$

11. If r = $\frac{1}{2}$ and n=4, then G.P. series will be?

$a_{1},\frac{a_{1}}{2},\frac{a_{1}}{4},\frac{a_{1}}{8}$

$a_{1},\frac{a_{1}}{2},\frac{a_{1}}{4},\frac{a_{1}}{8}$

78.85%

$a_{1},\frac{-a_{1}}{2},\frac{-a_{1}}{4},\frac{-a_{1}}{8}$

$a_{1},\frac{-a_{1}}{2},\frac{-a_{1}}{4},\frac{-a_{1}}{8}$

5.77%

$a_{1},\frac{a_{1}}{4},\frac{a_{1}}{8},\frac{a_{1}}{16}$

$a_{1},\frac{a_{1}}{4},\frac{a_{1}}{8},\frac{a_{1}}{16}$

9.62%

none of the above

none of the above

5.77%

$a_{n} = a\times r^{n-1}$

here, r = $\frac{1}{2}$ and n=4

so the series will be:

$a_{1},\frac{a_{1}}{2},\frac{a_{1}}{4},\frac{a_{1}}{8}$

12. Find the sum of the geometric series 3 + 9 + 27 + 81 + . . . where there are 5 terms in the series.

363

363

77.55%

362

362

6.12%

242

242

2.04%

243

243

14.29%

For this series, we have a = 3, r = 3 and n = 5.

$s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{5} = 3 \times \frac{3^{5} -1}{3-1} = 363$

13. Find the sum of the geometric series 8, − 4, + 2, − 1,. . . where there are 7 terms in the series.

5.37

5.37

78.57%

5.50

5.50

7.14%

6.34

6.34

4.76%

None of the above

None of the above

9.52%

For this series, we have a = 8, r = $\frac{-1}{2}$ and n = 7.

Thus

$s_{n} = a \times \frac{1-r^{n}}{1-r}$ if r<1

$s_{7} = 8 \times \frac{1-(\frac{-1}{2})^{7}}{1-(\frac{-1}{2})}$

S₇ =5.37.

14. How many terms are there in the geometric progression 4, 8, 16, . . ., 512?

5

5

2.22%

6

6

4.44%

7

7

8.89%

8

8

84.44%

Here a = 4 and r = 2. nth term=512. But the formula for the nth term is $ar^{n-1}$

So

512 = $4\times 2^{n-1}$

128= $2^{n-1}$

$2^{7} = 2^{n-1}$

7 = n − 1

n = 8.

15. If the common ratio in a specific series is 3. Last term is 486 and sum of the terms is 728. Then find out the first term of gp?

3

3

5%

1

1

7.5%

2

2

82.5%

8

8

5%

The series of gp is: 1, ar², ar³,………

Common ratio=3

Last term= 486

So $a_{n} = a\times r^{n-1}$ = 486

$a_{n} = a\times 3^{n-1}$=486

a(3ⁿ) = $486\times 3$ = 1458-----(1)

Now, sum of G.P.=

=  728

By putting the value of r and using eq(1) we get

$\frac{1458-a}{2} = 728$

1458 - a = 1456

So a= 2.

16. How do you find S₁₀ for the geometric series 1 + 5 + 25 + 125 +…?

25,00,000

25,00,000

7.69%

23,87,463

23,87,463

7.69%

24,41,406

24,41,406

74.36%

None of the above

None of the above

10.26%

As we know first term a=1 and r=5.

$s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{10} = 1 \times \frac{5^{10} -1}{5-1}$

= 24,41,406

17. What is the sum of below given infinite g.p.?

$\frac{1}{2} +\frac{1}{3}+\frac{1}{4} +\frac{1}{9} +\frac{1}{8} +\frac{1}{27}+$…………………..=

$\frac{1}{2}$

$\frac{1}{2}$

10%

$\frac{2}{3}$

$\frac{2}{3}$

5%

$\frac{2}{5}$

$\frac{2}{5}$

7.5%

$\frac{3}{2}$

$\frac{3}{2}$

77.5%

18. How do you find $s_{17}$ for the geometric series 5 + 15 + 45 + 135 + …?

32,28,50,405

32,28,50,405

73.68%

12,91,40,631

12,91,40,631

7.89%

34,2154,214

34,2154,214

13.16%

None of the above

None of the above

5.26%

As we know first term a=5 and r=3.

Formula= $s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{17} = 5 \times \frac{3^{17} -1}{3-1}$

=32,28,50,405

19. How do you find the sum of the first 6 terms of the geometric series: 6+ 36 + 216…?

33454

33454

7.69%

55986

55986

79.49%

43659

43659

5.13%

76463

76463

7.69%

Common ratio = 6 and  First term = 6

As per the formula:

$s_{n} = a \times \frac{r^{n} -1}{r-1}$ if r>1

$s_{6} = 6 \times \frac{6^{6} -1}{6-1}$

= 55986

20. Find out the sum of the next infinite geometric series, if it exists? 3 + 2.25 + 1.6875 + 1.265625+…?

10

10

4.76%

11

11

7.14%

12

12

83.33%

13

13

4.76%

Here a=3

r=0.75

Since |r|<1

So = $\frac{a_{1}}{1-r}$

Or, sum= $\frac{3}{(1-0.75)} = 12$

21. What is the sum of the first 10 terms of a geometric series with $a_{1} = 20$ and r  =$\frac{1}{5}$?

0.99

0.99

9.76%

24.99

24.99

78.05%

35

35

2.44%

17.44

17.44

9.76%

First term $a_{1} = 20$ , common ratio, r = $\frac{1}{5}$ = 0.2

Number of terms , n=10

Sum of first 10 terms is

$s_{n} = a \times \frac{1-r^{n}}{1-r}$ if r<1 $s_{10} = 20 \times \frac{1-0.2^{10}}{1-0.2}$ Sum of first 10 terms is 24.99.

22. Geometric Mean of three numbers a,b,c is

$(abc)^{\frac{1}{3}}$

$(abc)^{\frac{1}{3}}$

62.5%

$(abc)^{\frac{1}{2}}$

$(abc)^{\frac{1}{2}}$

22.92%

$(ac)^{\frac{1}{3}}$

$(ac)^{\frac{1}{3}}$

10.42%

None

None

4.17%

$(abc)^{\frac{1}{3}}$

23. The total of second and third terms in a GP is 12, and 60 for 3rd and 4th terms. Find the common ratio?

r = either 5 or -1

r = either 5 or -1

73.91%

r = either 6 or 1

r = either 6 or 1

10.87%

r = either -5 or 2

r = either -5 or 2

4.35%

None of the above

None of the above

10.87%

Here series will be like: a, ar, $ar^{2}$ , $ar^{3}$

or, ar+ar²=12 and ar²+ar³=60

or,$\frac{ar^{2}+ar^{3}}{ar+ar^{2}} = 5$

or r+r² = 5(1+r)

r²+r = 5r+5

r²-4r-5 = 0

(r+1)(r-5) = 0

So r = either 5 or -1

24. The 9th term of a GP is 16 times the 5th term. What will be the first term when its seventh term is 96.

1

1

6.82%

2

2

9.09%

2.5

2.5

6.82%

1.5

1.5

77.27%

As we know the formula of nth term of GP = arn-1
9th term = ar8 and 5th term =ar4
Given that
ar8 = 16ar4
=> r4 = 16
=> r = 2
Given ar6 = 96
put value of r above
a(2)6 = 96
a = $\frac{96}{64} = 1.5$
Hence, first term of the GP is 1.5.

25. If a is 30 and b is 45. Find out the G.M of ab is?

30.50

30.50

10.64%

28.62

28.62

2.13%

36.74

36.74

78.72%

22.56

22.56

8.51%

As we know: in geometric mean problems ab = $\sqrt{ab}$

Here a = 30, b = 45

G.M. of ab = $\sqrt{ab}$ = 36.74

26. What is the 9th term of the geometric sequence 4, 12, 36,...?

6561

6561

4.88%

26244

26244

85.37%

19683

19683

4.88%

None of the above

None of the above

4.88%

The nth term of a geometric sequence is $a_{n} = a\times r^{n-1}$

Here a is the first term and r the common ratio

r = $\frac{a_{2}}{a_{1}} = \frac{a_{3}}{a_{2}} =......=\frac{a_{n}}{a_{n-1}}$

Now a=4 and r = $\frac{12}{4} = 3$

So = $a_{9} = 4\times 3^{8} = 26244$

27. Let the sequence be 4, 24, 144, 864,……… then this sequence is

arithmetic sequence

arithmetic sequence

4.26%

Geometric progression

Geometric progression

89.36%

harmonic sequence

harmonic sequence

2.13%

None of the mentioned

None of the mentioned

4.26%

The ratio of each term with earlier term is same.

So , Geometric Progression

28. If in a specific Geometric Progression series n is an integer then which series will have common ratio 3?

g(n) = $6(3^{n-1})$

g(n) = $6(3^{n-1})$

60.87%

g(n) = $3n^{2}-1$

g(n) = $3n^{2}-1$

19.57%

g(n) = $2n^{2}+3n-1$

g(n) = $2n^{2}+3n-1$

8.7%

g(n) = $2n+ 3n^{2}$

g(n) = $2n+ 3n^{2}$

10.87%

In g(n) = $6(3^{n-1})$ Geometric Progression series as it consists coefficient of $3^{n-1}$

29. If Product of 3 successive terms of Geometric Progression is 8 then mid of those 3 successive terms will be

2

2

83.33%

12

12

10.42%

4

4

4.17%

13

13

2.08%

Let $\frac{x}{r}, x , xr$ be three terms ,then $\frac{x}{r} \times x \times xr = 8$

$x = 2$

30. If x, y, z are in GP then relation between x, y, z will be

2y = 2x + 3z

2y = 2x + 3z

13.73%

2x = y+z

2x = y+z

11.76%

y = $(xz)^{\frac{1}{2}}$

y = $(xz)^{\frac{1}{2}}$

50.98%

2z = x + z

2z = x + z

23.53%

The term y should be the Geometric Mean of term x and z.

×