Question:
Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98/25. Then the sum of the first 21 terms of an AP, whose first term is 10ar, nth term is an and the common difference is 10ar2, is equal to
Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98/25. Then the sum of the first 21 terms of an AP, whose first term is 10ar, nth term is an and the common difference is 10ar2, is equal to
Updated On: Jun 29, 2025
- 21 a11
- 22 a11
- 15 a16
- 14 a16
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
The correct answer is (A) : 21 a11
Let first term of G.P. be a and common ratio is r
Then, \(\frac{α}{1-r} = 5....(i)\)
\(α =\frac{(r^5-1)}{(r-1)}\)
\(= \frac{98}{25}\)
\(⇒ 1-r^5 = \frac{98}{125}\)
\(∴ r^5 = \frac{27}{125}\)
\(r = (\frac{3}{5})^{\frac{3}{5}}\)
∴ Then, \(S_{21} = \frac{21}{2}[2×10ar + 20×10ar^2]\)
\(= 21[10ar+10.10ar^2]\)
\(= ^{21}α_{11}\)
Was this answer helpful?
3
1
Top Questions on Geometric Progression
- If \( a, b, c \) elements are in geometric progression, then \( \log a, \log b, \log c \) will be in:
- TG EDCET - 2025
- Mathematics
- Geometric Progression
- Find the sum of the infinite geometric series: $$ S = 8 + 4 + 2 + \cdots $$ if it converges.
- BITSAT - 2025
- Mathematics
- Geometric Progression
- If $ a, a r, a r^2 $ are in a geometric progression (G.P.), then find the value of: $ \left| a \quad a r \right| \quad \left| a r^2 \quad a r^3 \right| = \left| a r^3 \quad a r^6 \right| $
- KEAM - 2025
- Mathematics
- Geometric Progression
- The product of the first five terms of a GP is 32. What is the 3rd term?
- KEAM - 2025
- Mathematics
- Geometric Progression
- If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is:
- JEE Main - 2025
- Mathematics
- Geometric Progression
View More Questions
Questions Asked in JEE Main exam
- Let \( f(x) = \frac{x - 5}{x^2 - 3x + 2} \). If the range of \( f(x) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^2 \) equals to.
- JEE Main - 2025
- Functions
- CrCl\(_3\).xNH\(_3\) can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of 0.558°C. Assuming 100\% ionization of this complex and coordination number of Cr is 6, the complex will be:
- JEE Main - 2025
- Colligative Properties
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Complex numbers
- Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
- JEE Main - 2025
- Differential Equations
View More Questions
Concepts Used:
Geometric Progression
What is Geometric Sequence?
A geometric progression is the sequence, in which each term is varied by another by a common ratio. The next term of the sequence is produced when we multiply a constant to the previous term. It is represented by: a, ar1, ar2, ar3, ar4, and so on.
Properties of Geometric Progression (GP)
Important properties of GP are as follows:
- Three non-zero terms a, b, c are in GP if b2 = ac
- In a GP,
Three consecutive terms are as a/r, a, ar
Four consecutive terms are as a/r3, a/r, ar, ar3 - In a finite GP, the product of the terms equidistant from the beginning and the end term is the same that means, t1.tn = t2.tn-1 = t3.tn-2 = …..
- If each term of a GP is multiplied or divided by a non-zero constant, then the resulting sequence is also a GP with a common ratio
- The product and quotient of two GP’s is again a GP
- If each term of a GP is raised to power by the same non-zero quantity, the resultant sequence is also a GP.
If a1, a2, a3,… is a GP of positive terms then log a1, log a2, log a3,… is an AP (arithmetic progression) and vice versa