Geometric Progression Questions and Answers

Geometric progression Questions 

In this page Geometric Progression Questions and Answers is given for Your Practice . This will help you in Solving Questions of Geometric Progression.
Geometric Progressions 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.
Geometric Progression Question and Answers

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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 [latex]s_{10} = 20 \times \frac{1-0.2^{10}}{1-0.2} [/latex] 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.

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