Question
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$.$$\left(1+\frac{3}{1}\right)\left(1+\frac{5}{4}\right)\left(1+\frac{7}{9}\right) \ldots\left(1+\frac{(2 n+1)}{n^{2}}\right)=(n+1)^{2} .$$
Step 1
e., when $n=1$. The left-hand side (LHS) of the given equation becomes: $$ \left(1+\frac{3}{1}\right) = 1 + 3 = 4 $$ The right-hand side (RHS) when $n=1$ is: $$ (1+1)^{2} = 2^{2} = 4 $$ Since LHS = RHS, the base case is true. Show more…
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