Question

Which of the following is larger? 99⁵⁰ + 100⁵⁰  or 101⁵⁰

Answer 1

We have (101)⁵⁰ = (100 + 1)⁵⁰

= `100^50 + 50(100)^49 + (50*49)/(2*1) (100)^48 + (50*49*48)/(3*2*1) (100)^47 +` ......(1)

Similarly 99⁵⁰ = (100 – 1)⁵⁰

= `100^50 - 50 * 100^59 + (50*49)/(2*1) (100)^48 - (50*49*48)/(3*2*1) (100)^47 +`  ....(2)

Subtracting (2) from (1), we get

101⁵⁰ – 99⁵⁰ = `2  50*(100)^49 + (50*49*48)/(3*2*1) 100^47 +`  ....

⇒ 101⁵⁰ – 99⁵⁰ = `100^50 + 2  (50*49*48)/(3*2*1)  10^47 +`  ....

⇒ 101⁵⁰ – 99⁵⁰ > 100⁵⁰

Hence 101⁵⁰ > 99⁵⁰ + 100⁵⁰