For all n ≥ 1, prove that 1 + 1/(1+2) + 1/(1+2+3) + .........+ 1/(1+2+3+.....+n) =2n/(n+1)

Question

For all n ≥ 1, prove that

 \(1+ \frac{1}{(1+2)} + \frac{1}{(1+2+3)}\) + ........ + \(\frac{1}{(1+2+3+...+n)}\) = \(\frac{2n}{n+1}\)

Answer 1

 \(1+ \frac{1}{(1+2)} + \frac{1}{(1+2+3)}\) + ........ + \(\frac{1}{(1+2+3+...+n)}\) = \(\frac{2n}{n+1}\)

Put n = 1

p(1) = 1 + \(\frac{2}{(1+1)} = I \) which is true

Assuming that true for p(k)

p(k): \(1+ \frac{1}{(1+2)} + \frac{1}{(1+2+3)}\) + ...... + \(\frac{1}{(1+2+3+...+k)}\)

= \(\frac{2k}{(k+1)}\).

Let p(k + 1):

Hence by using the principle of mathematical induction true for all n ∈ N.