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
- 21 a11
- 22 a11
- 15 a16
- 14 a16
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, \(S_{21} = \frac{21}{2}[2×10ar + 20×10ar^2]\)
\(= 21[10ar+10.10ar^2]\)
\(= ^{21}α_{11}\)
Top JEE Main Mathematics Questions
- Let $S = \{\theta \in (-2\pi, 2\pi) : \cos\theta + 1 = \sqrt{3} \sin\theta\}$. Then $\sum_{\theta \in S} \theta$ is equal to:
- JEE Main - 2026
- Mathematics
- Trigonometry
If for \( 3 \leq r \leq 30 \), \[ \binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}, \] then \( m \) equals: ________
- JEE Main - 2026
- Mathematics
- Combinatorics
- The number of points in the interval \( [2, 4] \), at which the function \( f(x) = \left\lfloor x^2 - x - \frac{1}{2} \right\rfloor \), where \( \left\lfloor \cdot \right\rfloor \) denotes the greatest integer function, is discontinuous, is _______.
Let \[ \alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots \infty \] and \[ \beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \infty. \]
Then the value of \[ (0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_{5}(\beta)} \] is equal to: ________- JEE Main - 2026
- Mathematics
- Sequences and Series
Let \( y = y(x) \) be the solution of the differential equation:
\[ \frac{dy}{dx} + \left( \frac{6x^2 + (3x^2 + 2x^3 + 4)e^{-2x}}{(x^3 + 2)(2 + e^{-2x})} \right)y = 2 + e^{-2x}, \quad x \in (-1, 2) \]
satisfying \( y(0) = \frac{3}{2} \).
If \( y(1) = \alpha \left(2 + e^{-2}\right) \), then the value of \( \alpha \) is ________.- JEE Main - 2026
- Mathematics
- Differential Equations
Top JEE Main Geometric Progression Questions
- If \( a_1, a_2, a_3, \ldots \) are in increasing geometric progression such that \[ a_1 + a_3 + a_5 = 21, a_1 a_3 a_5 = 64, \] then \( a_1 + a_2 + a_3 \) is:
- JEE Main - 2026
- 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
- If in a G.P. of64terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:
- JEE Main - 2024
- Mathematics
- Geometric Progression
- Let \( a \) and \( b \) be two distinct positive real numbers. Let the 11th term of a GP, whose first term is \( a \) and third term is \( b \), be equal to the \( p \)-th term of another GP, whose first term is \( a \) and fifth term is \( b \). Then \( p \) is equal to
- JEE Main - 2024
- Mathematics
- Geometric Progression
- If the range of $f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}, \, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to ____.
- JEE Main - 2024
- Mathematics
- Geometric Progression
Top JEE Main Questions
- 1 mole of ideal diatomic gas is enclosed in a cylinder piston arrangement having cross-sectional area of piston \(4\,\text{cm}^2\). If gas has only rotational modes and \(P_{atm}=100\ \text{kPa}\), some amount of heat is added to the system as a result piston moves up slowly by \(2.5\,\text{cm}\). If temperature change is \(1.2^\circ C\). Find heat given to gas.
- JEE Main - 2026
- Thermal Physics
- Let $S = \{\theta \in (-2\pi, 2\pi) : \cos\theta + 1 = \sqrt{3} \sin\theta\}$. Then $\sum_{\theta \in S} \theta$ is equal to:
- JEE Main - 2026
- Trigonometry
Refer the figure below. \( \mu_1 \) and \( \mu_2 \) are refractive indices of air and lens material respectively. The height of image will be _____ cm.
- JEE Main - 2026
- Optics and Refraction
In single slit diffraction pattern, the wavelength of light used is \(628\) nm and slit width is \(0.2\) mm. The angular width of central maximum is \(\alpha \times 10^{-2}\) degrees. The value of \(\alpha\) is ____.
- JEE Main - 2026
- Interference and Diffraction
\(t_{100\%}\) is the time required for 100% completion of a reaction, while \(t_{1/2}\) is the time required for 50% completion of the reaction. Which of the following correctly represents the relation between \(t_{100\%}\) and \(t_{1/2}\) for zero order and first order reactions respectively
- JEE Main - 2026
- Zero-order Reactions
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