Question

Prove the following by using the principle of mathematical induction for all $n in N$:$(1+3/1)(1+5/4)(1+7/9)...(1+((2n+1))/(n^2))=(n+1)^2$

Answer 1

$P(k):(1+3)(1+5/4)(1+7/9)…{1+((2k+1))/(k^(2))}=(k+1)^(2)$.
Now, $(1+3)(1+5/4)…(1+(2k+1)/k^(2)){1+(2(k+1)+1)/((k+1)^(2))}$
$ = (k+1)^(2) xx ((k+1)^(2)+(2k+3))/((k+1)^(2)) = (k^(2)+3k+4) = (k+2)^(2)$.