Let a ,   b and c be in G . P . with common ratio r , where a ≠ 0 and 0 < r ≤ 1 2 . If 3 a ,   7 b and 15 c are the first three terms of an A . P . , then the 4 t h term of this A . P . is :

Step-by-step solution

We know that, the n t h term of a G . P . with first term a and common ratio r is t n = a r n - 1 . Then, b = a r and c = a r 2 Now, 3 a ,   7 b and 15 c are in A . P . ⇒ 14 b = 3 a + 15 c ⇒ 14 a r = 3 a + 15 ( a r 2 ) ⇒ 14 r = 3 + 15 r 2 ⇒ 15 r 2 - 14 r + 3 = 0 ⇒ 3 r - 1 5 r - 3 = 0 ⇒ r = 1 3 or   r = 3 5 The only acceptable value is r = 1 3 , because r ∈ 0 ,   1 2 ∴ the first term of the A . P . is A = 3 a and the common difference is d = 7 b - 3 a =