If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then write the value of k.

Question

Philoid · Accepted Answer

we know

……(1)

Given 2 + 4 + 6 + 8…….2n=k(1 + 3 + 5…….2n - 1)

From (1) n(n + 1)=k(1 + 3 + 5…2n - 1) …………….(2)

……. (3)

Thus substituting the values from (1) in (3), we get,

1 + 3 + 5…(2n - 1) = n(2n + 1) - n(n + 1)

1 + 3 + 5…(2n - 1) = n² …………….(4)

Substituting (4) in (2), we get,

n(n + 1) = k(n²)

k = (n + 1)/n