The sum of infinitely many terms of a decreasing GP is equal to the greatest value of the function $f (x) = x^{3} + 3 x - 9$ in the interval $[- 2, 3]$ , and the difference between the first two terms is $f' (0)$ . Then the common ratio of the GP is

Answer (Detailed Solution Below)View answer ▾

Answer: Option C — $\frac{2}{3}$

Detailed Solution

$f' (x) = 3 x^{2} + 3 > 0$ , so f is increasing and its greatest value is $f (3) = 27 + 9 - 9 = 27$ . Also $f' (0) = 3$ .

$\frac{a}{1 - r} = 27$ and $a - a r = a (1 - r) = 3$ . Multiplying, $a^{2} = 81$ , so $a = 9$ (a decreasing GP with sum 27 has positive terms).

$1 - r = \frac{3}{9} = \frac{1}{3}$ , so $r = \frac{2}{3}$ .

✓ Check

The GP 9, 6, 4, ... has sum

\frac{9}{1/3} = 27

, and

9 - 6 = 3

⚠ Common trap

(B)

\frac{4}{3}

comes from

a = - 9

, which gives

1 - r = - \frac{1}{3}

; but then

∣ r ∣ > 1

and the infinite sum does not exist.

Quick tip

$\frac{a}{1 - r} = 27$ and $a (1 - r) = 3 \Rightarrow a = 9$ , $r = \frac{2}{3}$

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Arithmetic & Geometric Progression — Preparation

This question tests your understanding of Arithmetic & Geometric Progression, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

An arithmetic progression (AP) is a sequence where each term differs from the previous one by a constant amount, the common difference. A geometric progression (GP) is a sequence where each term is the previous one multiplied by a constant, the common ratio. This topic covers the nth term and sum of n terms of an AP and GP, the sum of an infinite GP, arithmetic and geometric means, and word problems. Key formulas are the AP nth term a + (n - 1)d, the AP sum n/2 [2a + (n - 1)d], the GP nth term ar^(n-1), and the infinite GP sum a/(1 - r) for |r| < 1. A useful trick is to choose symmetric terms, such as a - d, a, a + d for three terms of an AP, which simplifies many questions. Practise the AP and GP questions below, each with a step-by-step solution.

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The sum of infinitely many terms of a decreasing GP is equal to the greatest value of the function f(x)=x3+3x−9 in the interval [−2,3], and the difference between the first two terms is f′(0). Then the common ratio of the GP is

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Arithmetic & Geometric Progression — Preparation

The sum of infinitely many terms of a decreasing GP is equal to the greatest value of the function $f (x) = x^{3} + 3 x - 9$ in the interval $[- 2, 3]$ , and the difference between the first two terms is $f' (0)$ . Then the common ratio of the GP is