Let b i > 1 for i = 1 , 2 , … . , 101 . Suppose log e ⁡ b 1 , log e ⁡ b 2 , … . . , log e ⁡ b 101 are in Arithmetic Progression (A.P.) with the common difference log e ⁡ 2 . Suppose a 1 , a 2 , … . , a 101 are in A.P. such that a 1 = b 1 and a 51 = b 51 . If t = b 1 + b 2 + … + b 51 and s = a 1 + a 2 + … + a 51 then

Step-by-step solution

If log e ⁡ b 1 , log e ⁡ b 2 … . log e ⁡ b 101 → A P ; difference(d) = log e ⁡ 2 ⇒ b 1 , b 2 , b 3 ....... b 101 → G P ; r = 2 ∴ b 1 , 2 b 1 , 2 2 b 1 ......... , 2 100 b 1 → G P a 1 , a 2 , a 3 .......... a 101 → A P Let Common Difference = D Given, a 1 = b 1 and a 51 = b 51 ⇒ a 1 + 50 D = 2 50 b 1 ∴ a 1 + 50 D = 2 50 a 1 ( A s b 1 = a 1 ) ⇒ D = 2 50 a 1 − a 1 50 t = b 1 + b 2 + ....... b 51 = b 1 + 2 b 1 + 2 2 b 1 + .......2 50 b 1 ⇒ t = b 1 ( 2 51 − 1 ) ; s = a 1 + a 2 + ....... a 51 ⇒ s = 51 2 ( 2 a 1 + 50 D )