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

Prime Course Trailer

Related Banners

Get PrepInsta Prime & get Access to all 200+ courses offered by PrepInsta in One Subscription

Also Check Out

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

6,94,765

6,94,765

11.57%

5,97,871

5,97,871

59.26%

2,44,406

2,44,406

6.02%

None of the above

None of the above

23.15%

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

69.46%

362

362

8.87%

242

242

3.94%

243

243

17.73%

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

69.23%

5.50

5.50

10.9%

6.34

6.34

6.41%

None of the above

None of the above

13.46%

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

8.98%

6

6

7.19%

7

7

16.17%

8

8

67.66%

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

13.49%

1

1

12.7%

2

2

66.67%

8

8

7.14%

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

10.95%

23,87,463

23,87,463

8.76%

24,41,406

24,41,406

67.88%

None of the above

None of the above

12.41%

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}

5.74%

\frac{2}{3}

\frac{2}{3}

12.3%

\frac{2}{5}

\frac{2}{5}

5.74%

\frac{3}{2}

\frac{3}{2}

76.23%

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

32,28,50,405

32,28,50,405

68.57%

12,91,40,631

12,91,40,631

11.43%

34,2154,214

34,2154,214

10.48%

None of the above

None of the above

9.52%

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

4%

55986

55986

86%

43659

43659

5%

76463

76463

5%

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

7.92%

11

11

5.94%

12

12

83.17%

13

13

2.97%

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}

82.61%

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}

5.22%

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}

9.57%

none of the above

none of the above

2.61%

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

78.85%

362

362

6.73%

242

242

1.92%

243

243

12.5%

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

83.33%

5.50

5.50

7.14%

6.34

6.34

3.57%

None of the above

None of the above

5.95%

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

2.04%

6

6

6.12%

7

7

7.14%

8

8

84.69%

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

4.65%

1

1

4.65%

2

2

77.91%

8

8

12.79%

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

6.49%

23,87,463

23,87,463

9.09%

24,41,406

24,41,406

76.62%

None of the above

None of the above

7.79%

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}

7.89%

\frac{2}{3}

\frac{2}{3}

6.58%

\frac{2}{5}

\frac{2}{5}

3.95%

\frac{3}{2}

\frac{3}{2}

81.58%

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

32,28,50,405

32,28,50,405

79.1%

12,91,40,631

12,91,40,631

5.97%

34,2154,214

34,2154,214

10.45%

None of the above

None of the above

4.48%

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

5.63%

55986

55986

83.1%

43659

43659

7.04%

76463

76463

4.23%

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

6.85%

11

11

5.48%

12

12

83.56%

13

13

4.11%

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

8%

24.99

24.99

81.33%

35

35

4%

17.44

17.44

6.67%

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}}

60.58%

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

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

23.08%

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

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

10.58%

None

None

5.77%

(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

78.49%

r = either 6 or 1

r = either 6 or 1

9.68%

r = either -5 or 2

r = either -5 or 2

3.23%

None of the above

None of the above

8.6%

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

5.05%

2

2

14.14%

2.5

2.5

5.05%

1.5

1.5

75.76%

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

76.6%

22.56

22.56

10.64%

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

2.5%

26244

26244

87.5%

19683

19683

3.75%

None of the above

None of the above

6.25%

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

2.94%

Geometric progression

Geometric progression

92.16%

harmonic sequence

harmonic sequence

1.96%

None of the mentioned

None of the mentioned

2.94%

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})

72.22%

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

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

13.33%

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

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

6.67%

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

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

7.78%

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

84.47%

12

12

9.71%

4

4

4.85%

13

13

0.97%

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

11.57%

2x = y+z

2x = y+z

7.44%

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

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

68.6%

2z = x + z

2z = x + z

12.4%

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

×

Please login to report

 

Also Check Out 

Get over 200+ course One Subscription

Courses like AI/ML, Cloud Computing, Ethical Hacking, C, C++, Java, Python, DSA (All Languages), Competitive Coding (All Languages), TCS, Infosys, Wipro, Amazon, DBMS, SQL and others

Checkout list of all the video courses in PrepInsta Prime Subscription

Checkout list of all the video courses in PrepInsta Prime Subscription