Let a, b and c be in GP with common ratio r, where a ≠ 0 and 0 < r ≤ 1/2. If 3a, 7b and 15c are the first three terms of an AP, then the 4th term of this AP is

Let a, b and c be in GP with common ratio r, where a ≠ 0… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 10 Apr · Shift II

Let

a

b

and

c

be in GP with common ratio

r

, where

a \neq 0

and

0 < r \leq \frac{1}{2}

. If

3a

7b

and

15c

are the first three terms of an AP, then the 4th term of this AP is

A.
$5a$
B.
$\frac{2}{3}a$
C.
$a$
D.
$\frac{7}{3}a$

Show answer and solution

Correct option: C

Solution

Since a, b, c are in GP with ratio r: b = ar, c = ar².

For 3a, 7b, 15c in AP: 2(7b) = 3a + 15c → 14ar = 3a + 15ar² → 15r² - 14r + 3 = 0 → (3r-1)(5r-3) = 0, giving r = 1/3 (since r ≤ 1/2).

Common difference d = 7b - 3a = 7ar - 3a = 7a(1/3) - 3a = (7/3 - 3)a = -2a/3.

4th term = 15c + d = 15ar² + (-2a/3) = 15a(1/9) - 2a/3 = 5a/3 - 2a/3 = a.

[JEE Main 2019 · 10 Apr · Shift II]

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