Geometric Progressions MCQ Quiz - Objective Question with Answer for Geometric Progressions - Download Free PDF
Last updated on May 20, 2025
Latest Geometric Progressions MCQ Objective Questions
Geometric Progressions Question 1:
Let the geometric mean of 27, 60, 108, 150 and 225 be x, then, what is the harmonic mean of x and 60?
Answer (Detailed Solution Below)
Geometric Progressions Question 1 Detailed Solution
Given:
The numbers are 27, 60, 108, 150, and 225, and the geometric mean is denoted by x. We need to find the harmonic mean of x and 60.
Formula used:
Geometric mean (x) = (Product of numbers)^(1/n), where n is the number of terms, and the numbers are the given values.
Harmonic mean (H) of two numbers a and b is given by:
H = 2ab / (a + b)
Calculations:
Find the geometric mean (x) of the numbers 27, 60, 108, 150, and 225.
x = (27 × 60 × 108 × 150 × 225)(1/5)
x = (3 × 3 × 3 × 3 × 2 × 2 × 5 × 3 × 3 × 3 × 2 × 2 × 3 × 5 × 2 × 5 × 3 × 5 × 3 × 5)(1/5)
x = (310 × 25 × 55)(1/5)
x = 32 × 2 × 5
x = 90
Find the harmonic mean of x and 60.
Now that we know x = 90, we can find the harmonic mean of x and 60:
H = 2 × 90 × 60 / (90 + 60)
H = 10800 / 150
H = 72
∴ The harmonic mean of x and 60 is 72.
Geometric Progressions Question 2:
If f is a function satisfying for all such that and \(\sum_{x=1}^{n}\) f(x) = 120 , then the value of n is:
Answer (Detailed Solution Below)
Geometric Progressions Question 2 Detailed Solution
Concept:
Geometric Progression (G.P.): A sequence where each term is obtained by multiplying the previous term by a constant called the common ratio.
Sum of n terms (Sn) of G.P. is given by: \( S_n = \frac{a(r^n - 1)}{r - 1} \)
- a: First term of the G.P.
- r: Common ratio of the G.P.
- n: Number of terms in the series
Calculation:
Given, f(1) = 3
Using the property: \( f(x + y) = f(x) \times f(y) \)
⇒ f(2) = f(1 + 1) = f(1) × f(1) = 3 × 3 = 9
⇒ f(3) = f(1 + 2) = f(1) × f(2) = 3 × 9 = 27
⇒ f(4) = f(1 + 3) = f(1) × f(3) = 3 × 27 = 81
So, \( f(1), f(2), f(3), \dots \) = 3, 9, 27, 81, … forms a G.P. with a = 3 and r = 3
Given, \( \sum_{x=1}^{n} f(x) = 120 \)
⇒ \( S_n = \frac{3(3^n - 1)}{3 - 1} \)
⇒ \( 120 = \frac{3(3^n - 1)}{2} \)
⇒ \( 120 = \frac{3}{2}(3^n - 1) \)
⇒ \( 240 = 3(3^n - 1) \)
⇒ \( 80 = 3^n - 1 \)
⇒ \( 3^n = 81 \)
⇒ \( 3^n = 3^4 \Rightarrow n = 4 \)
∴ The value of n is 4.
Geometric Progressions Question 3:
The third term of GP is 4. Find the product of its first 5 terms ?
Answer (Detailed Solution Below)
Geometric Progressions Question 3 Detailed Solution
Concept:
Let us consider sequence a1, a2, a3 …. an is a G.P.
- Common ratio = r = \(\frac{{{{\rm{a}}_2}}}{{{{\rm{a}}_1}}} = \frac{{{{\rm{a}}_3}}}{{{{\rm{a}}_2}}} = \ldots = \frac{{{{\rm{a}}_{\rm{n}}}}}{{{{\rm{a}}_{{\rm{n}} - 1}}}}\)
- nth term of the G.P. is an = arn−1
Calculation:
Given: Third term of GP = 4
So, a3 = ar2 = 4
Now, product of five time term is given by = a × ar × ar2 × ar3 ×ar4
= a5r10
= (ar2)5
= 45
Geometric Progressions Question 4:
A geometric progression has first term 3 and the sum of all its 2n terms is three times the sum of its odd place terms, the common ratio is -
Answer (Detailed Solution Below)
Geometric Progressions Question 4 Detailed Solution
Explanation:
Given a = 3
Let common ratio be r
Then GP is a, ar, ar2, ar3, ....(2n th term)
Then for the GP taking odd places term is
a, ar2, ar4, ... (n the term)
i.e, common ratio is r2
By given condition
S2n = 3 × Sn
\({a(r^{2n}-1)\over r-1}=3\times {a({(r^2)}^n-1)\over r^2-1}\)
\({(r^{2n}-1)\over r-1}=3\times {({r}^{2n}-1)\over r^2-1}\)
\(1=\frac3{r+1}\)
r + 1 = 3
r = 2
Common ratio of the given GP is 2
Option (1) is true.
Geometric Progressions Question 5:
Let p = ln(x), q = ln(x3) and r = ln(x5), where x > 1. Which of the following statements is/are correct?
I. p, q and r are in AP.
Il. p, q and r can never be in GP.
Select the answer using the code given below.
Answer (Detailed Solution Below)
Geometric Progressions Question 5 Detailed Solution
Concept:
Arithmetic Progression (AP):
- The terms are in AP if the difference between consecutive terms is constant.
- The common difference is given by: \(d = q - p = r - q\)
Geometric Progression (GP):
- The terms are in GP if the ratio between consecutive terms is constant.
- The common ratio is given by: \(r = q \times p\)
Calculation:
Given
p = ln(x),
q = ln(x3) = 3 lnx
and r = ln(x5) = 5 lnx
Clearly, p - q = r - q = \(2lnx\)
⇒ p, q, r are in AP
Also \(\frac{q}{p} ≠ \frac{r}{p} \because \frac{q}{p} = 3 , \frac{r}{p} = \frac{5}{3}\)
∴ p, q, r can never be in GP
∴ Option (c) is correct.
Top Geometric Progressions MCQ Objective Questions
The third term of a G.P. is 9. The product of its first five terms is
Answer (Detailed Solution Below)
Geometric Progressions Question 6 Detailed Solution
Download Solution PDFConcept:
Five terms in a geometric progression:
If a G.P. has first term a and common ratio r then the five consecutive terms in the GP are of the form \(\rm \dfrac{a}{r^2},\dfrac{a}{r},a,ar,ar^2\) .
Calculation:
Let us consider a general geometric progression with common ratio r.
Assume that the five terms in the GP are \(\rm \dfrac{a}{r^2},\dfrac{a}{r},a,ar,ar^2\).
It is given that third term is 9.
Therefore, a = 9.
Now the product of the five terms is given as follows:
\(\rm \dfrac{a}{r^2}\times\dfrac{a}{r}\times a\times ar \times ar^2 = a^5\)
But we know that a = 9.
Thus, the product is \(9^5=3^{10}\).
If the sum of n numbers in the GP 5, 10, 20, ... is 1275 then n is ?
Answer (Detailed Solution Below)
Geometric Progressions Question 7 Detailed Solution
Download Solution PDFConcept:
Let us consider sequence a1, a2, a3 …. an is a G.P.- Common ratio = r = \(\frac{{{{\rm{a}}_2}}}{{{{\rm{a}}_1}}} = \frac{{{{\rm{a}}_3}}}{{{{\rm{a}}_2}}} = \ldots = \frac{{{{\rm{a}}_{\rm{n}}}}}{{{{\rm{a}}_{{\rm{n}} - 1}}}}\)
- nth term of the G.P. is an = arn−1
- Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - {\rm{\;}}1}}\); where r >1
- Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {1 - {\rm{\;}}{{\rm{r}}^{\rm{n}}}} \right)}}{{1 - {\rm{\;r}}}}\); where r <1
- Sum of infinite GP = \({{\rm{s}}_\infty } = {\rm{\;}}\frac{{\rm{a}}}{{1{\rm{\;}} - {\rm{\;r}}}}{\rm{\;}}\) ; |r| < 1
Calculation:
Given series is 5, 10, 20, ...
Here, a = 5, r = 2
Sum of n numbers = sn = 1275
To Find: nAs we know that, Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - {\rm{\;}}1}}\); where r >1
∴ sn = \(\frac{{{\rm{5\;}}\left( {{{\rm{2}}^{\rm{n}}} - 1} \right)}}{{{\rm{2}} - {\rm{\;}}1}}\)
1275 = 5 × (2n - 1)
⇒ 255 = (2n - 1)
⇒ 2n = 256
⇒ 2n = 28
∴ n = 8
For what possible value of x are the numbers - 2/7, x, - 7/2 are in a GP ?
Answer (Detailed Solution Below)
Geometric Progressions Question 8 Detailed Solution
Download Solution PDFCONCEPT:
If a, b and c are in a GP then b2 = ac
CALCULATION:
Given: The numbers - 2/7, x, - 7/2 are in GP
As we know that, if a, b and c are in GP then b2 = ac
Here, a = - 2/7, b = x and c = - 7/2
⇒ x2 = (-2/7) × (-7/2) = 1
⇒ x = ± 1
Hence, correct option is 3.
In a G.P. of positive terms , if every term is equal to the sum of next two terms. Then find the common ratio of the G.P.
Answer (Detailed Solution Below)
Geometric Progressions Question 9 Detailed Solution
Download Solution PDFConcept:
Let us consider sequence a1, a2, a3 …. an is a G.P.
- Common ratio = r = \(\frac{{{{\rm{a}}_2}}}{{{{\rm{a}}_1}}} = \frac{{{{\rm{a}}_3}}}{{{{\rm{a}}_2}}} = \ldots = \frac{{{{\rm{a}}_{\rm{n}}}}}{{{{\rm{a}}_{{\rm{n}} - 1}}}}\)
- nth term of the G.P. is an = arn−1
Sin18o = \(\frac{\sqrt{5}- 1}{4}\)
Calculation:
We know that if the first term of a G.P is 'a' and the common ratio is 'r' then in this case then G.P = a, ar, ar2............
Since we have given a = ar + ar2
Now, 1= r + r2
⇒ r2 + r - 1 = 0
After solving we get r = \(\frac{\sqrt{5}- 1}{2}\) = 2 ×\( \frac{\sqrt{5}- 1}{4}\) = 2 Sin18°
The value of \(\rm 9^{\tfrac{1}{3}} 9^{\tfrac{1}{9}} 9^{\tfrac{1}{27}}\ ...\ \infty\) is:
Answer (Detailed Solution Below)
Geometric Progressions Question 10 Detailed Solution
Download Solution PDFConcept:
Geometric Progression (GP):
- The series of numbers where the ratio of any two consecutive terms is the same is called a Geometric Progression.
- A Geometric Progression of n terms with first term a and common ratio r is represented as:
a, ar, ar2, ar3, ..., arn-2, arn-1.
- The sum of the first n terms of a GP is: Sn = \(\rm a\left(\dfrac{r^n-1}{r-1}\right)\).
- The sum to ∞ of a GP, when |r| < 1, is: S∞ = \(\rm \dfrac{a}{1-r}\).
Calculation:
Let us consider the infinite series \(\rm \dfrac{1}{3}+\dfrac{1}{9}+\dfrac{1}{27}+\ ... \infty\).
Here, a = \(\rm \dfrac{1}{3}\) and r = \(\rm \dfrac{\tfrac{1}{9}}{\tfrac{1}{3}}=\dfrac{1}{3}\).
∴ S∞ = \(\rm \dfrac{a}{1-r}=\dfrac{\tfrac{1}{3}}{1-\tfrac{1}{3}}=\dfrac{\tfrac{1}{3}}{\tfrac{2}{3}}=\dfrac{1}{2}\).
Now, let P = \(\rm 9^{\tfrac{1}{3}} 9^{\tfrac{1}{9}} 9^{\tfrac{1}{27}}\ ...\ \infty\).
∴ P = \(\rm 9^{\tfrac{1}{3}+\tfrac{1}{9}+\tfrac{1}{27}+\ ... \infty}=9^{\tfrac{1}{2}}=\sqrt9=3\).
A man has 2 parents, 4 grandparents, 8 great- grand parents, and so on . Find the number of ancestors during the 8 generations preceding his own .
Answer (Detailed Solution Below)
Geometric Progressions Question 11 Detailed Solution
Download Solution PDFConcept:
Let us consider sequence a1, a2, a3 …. an is an G.P.
- Sum of n terms = s = \(\rm \frac{a\left ( r^{n}-1 \right )}{r-1}\); where r >1
- Sum of n terms = s = \(\rm \frac{a\left ( 1-r^{n} \right )}{1-r}\) ; where r <1
Calculation:
The required no. of ancestors
= 2 + 4 + 8 +... upto 8 terms
As we know that sum of G.P ,
S = \(\rm \frac{a\left ( r^{n}-1 \right )}{r-1}\)
Where , a = 2, r = 2 and n = 8
⇒ No. of ancestors required = \(\rm \frac{2\times \left ( 2^{8}-1\right )}{2-1}\) = \(\rm 2 \times \left ( 2^{8} -1\right )\) = 510
∴ No. of ancestors required is 510.
The correct option is 4.
The third term of a GP is 3. What is the product of its first five terms?
Answer (Detailed Solution Below)
Geometric Progressions Question 12 Detailed Solution
Download Solution PDFConcept:
Let us consider sequence a1, a2, a3 …. an is a G.P.
Common ratio = r = \(\rm a_2\over a_1 \) = \(\rm a_3\over a_2 \) = \(\rm a_n\over a_{n-1} \)
Calculation:
Consider,
(a = 3) be the 3rd term of the G.P series,
So, we can write the five terms as,
\(\rm a\over r^2\), \(\rm a\over r\), a, ar, ar2
So, the product of the five terms (P) will be,
P = \(\rm a\over r^2\) × \(\rm a\over r\) × a × ar × ar2 = a5
Since,
a = 3,
∴ The product of the first five terms (P) = 35 = 243
For the series 1 + 3 + 32 + ... , the sum to n terms is 3280. Find the value of n.
Answer (Detailed Solution Below)
Geometric Progressions Question 13 Detailed Solution
Download Solution PDFConcept:
Geometric Progression (GP): The series of numbers where the ratio of any two consecutive terms is the same, is called a Geometric Progression.
- A Geometric Progression of n terms with first term a and common ratio r is represented as:
a, ar, ar2, ar3, ..., arn-2, arn-1.
- The sum of the first n terms of a GP is:
Sn = \(\frac{a(r^n-1)}{r-1}\) if r > 1 or Sn = \(\frac{a(1 -r^n)}{1-r}\) if r < 1
Calculation:
For the given geometric series 1 + 3 + 32 + ..., we have a = 1 and r = 3.
Let the sum of first n terms be equal to 3280.
∴ Sn = \(\rm 1\left(\frac{3^n\ -\ 1}{3\ -\ 1}\right)\) = 3280
⇒ \(\rm \left(\frac{3^n\ -\ 1}{2}\right)\) = 3280
⇒ 3n - 1 = 3280 × 2
⇒ 3n - 1 = 6560
⇒ 3n = 6561 = 38
⇒ n = 8.
If the sum of n numbers in the GP 4, 8, 16, ... is 2044 then n is ?
Answer (Detailed Solution Below)
Geometric Progressions Question 14 Detailed Solution
Download Solution PDFConcept:
Let us consider sequence a1, a2, a3 …. an is a G.P.- Common ratio = r = \(\frac{{{{\rm{a}}_2}}}{{{{\rm{a}}_1}}} = \frac{{{{\rm{a}}_3}}}{{{{\rm{a}}_2}}} = \ldots = \frac{{{{\rm{a}}_{\rm{n}}}}}{{{{\rm{a}}_{{\rm{n}} - 1}}}}\)
- nth term of the G.P. is an = arn−1
- Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - {\rm{\;}}1}}\); where r >1
- Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {1 - {\rm{\;}}{{\rm{r}}^{\rm{n}}}} \right)}}{{1 - {\rm{\;r}}}}\); where r <1
- Sum of infinite GP = \({{\rm{s}}_\infty } = {\rm{\;}}\frac{{\rm{a}}}{{1{\rm{\;}} - {\rm{\;r}}}}{\rm{\;}}\) ; |r| < 1
Calculation:
Given series is 4, 8, 16, ...
Here, a = 4, r = 2
Sum of n numbers = sn = 2044
To Find: nAs we know that, Sum of n terms of GP = sn = \(\frac{{{\rm{a\;}}\left( {{{\rm{r}}^{\rm{n}}} - 1} \right)}}{{{\rm{r}} - {\rm{\;}}1}}\); where r >1
∴ sn = \(\frac{{{\rm{4\;}}\left( {{{\rm{2}}^{\rm{n}}} - 1} \right)}}{{{\rm{2}} - {\rm{\;}}1}}\)
2044 = 4 × (2n - 1)
⇒ 511 = (2n - 1)
⇒ 2n = 512
⇒ 2n = 29
∴ n = 9
What is the geometric mean of 6, 8, 16 and 27?
Answer (Detailed Solution Below)
Geometric Progressions Question 15 Detailed Solution
Download Solution PDFConcept:
The geometric mean is defined as the nth root of the product of n numbers.
The geometric mean of a data set \(\rm {\textstyle \left\{a_{1},a_{2},\,\ldots ,\,a_{n}\right\}}\) is given by:
\(\rm GM = \rm {\textstyle \left\{a_{1}\times a_{2}\times\,\ldots \times\,a_{n}\right\}}^{\frac{1}{n}}\)
Calculation:
To Find: Geometric mean of 6, 8, 16 and 27
Here n = 4
Now,
\(\rm GM = \rm ({{6 \times 8 \times 16 \times 27}})^{\frac{1}{4}}\)
\(= \rm ({{2 \times 3 \times 2^3 \times 2^4 \times 3^3}})^{\frac{1}{4}} \\ =(2^8 \times 3^4)^{\frac{1}{4}}\\ = (2^2 \times 3)\\=12\)