Let a, b and c be in G.P. with common ratio r, where a 0 and 0 < r . If 3a, 7b and 15c are the first three terms of an A.P., then the 4th term of this A.P. is

Engineering

Mathematics

Geometric Progression and Its Properties

Arithmetic Progression and Its Properties

Question

Let a, b and c be in G.P. 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 A.P., then the 4^th term of this A.P. is

$\frac{2 a}{3}$

$\frac{7 a}{3}$

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b = ar

c =ar²

3a,7b and 15c are in A.P.

⇒ 14b = 3a + 15c

⇒ 14(ar) = 3a + 15ar²

⇒ 14r = 3 + 15r²

⇒ 15r² – 14r + 3 = 0 ⇒ (3r – 1)(5r – 3) = 0

$r = \frac{1}{3}, \frac{3}{5}$

Only acceptable value is $r = \frac{1}{3}, because r \in (0, \frac{1}{2}]$

$∴$ c.d = 7b – 3a = 7ar – 3a $= \frac{7}{3} a - 3 a = - \frac{2}{3} a$

$∴$ 4^th term $= 15 c - \frac{2}{3} a = \frac{15}{9} a - \frac{2}{3} a = a$