Let a 1 , a 2 , … , a 10 be an A . P . with common difference - 3 and b 1 , b 2 , … , b 10 be a G . P . with common ratio 2 . Let c k = a k + b k , k = 1 , 2 , … , 10 . If c 2 = 12 and c 3 = 13 , then ∑ k = 1 10 c k is equal to ______.

Step-by-step solution

C k = a k + b k where a k = a 1 + ( k - 1 ) ( - 3 ) and b k = b 1 2 k - 1 C k = a 1 + ( k - 1 ) ( - 3 ) + b 1 2 k - 1 ∑ k = 1 10 c k = ∑ k = 1 10 a k + ∑ k = 1 10 b k C 2 = a 1 - 3 + b 1 ( 2 ) = 12       ( 1 ) . . . . C 3 = a 1 - 6 + 4   b 1 = 13         ( 2 ) . . . . . . by solving (1) & (2) 2 b 1 - 3 = 1 ⇒ b 1 = 2 and a 1 = 11 ∑ k = 1 10 c k = 10 2 22 + 9 - 3 + 2 2 10 - 1 2 - 1 = - 25 + 2046 = 2021