Prove that...

Question

Prove that the sum of first p even numbers is (1+1/p) times the sum of first p odd numbers.

Answer 1

For the even numbers:

S_n = (n/2)(2(2) + (n - 1)2)

S_n = (n/2)(2(2) + 2n - 2)

S_n = (n/2)(4 + 2n - 2)

S_n = (n/2)(2 + 2n)

S_n = (1 + n)n

 

For the odd numbers

S_n = (n/2)(2(1) + (n - 1)2)
S_n = (n/2)(2(1) + 2n - 2)
S_n = (n/2)(1 + 2n - 2)
S_n = (n/2)(2n)
S_n = n²

 

Sum of evens / Sum of odds

(1 + n)n / n²

(1 + n) / n

1/n + 1