# Using the principle of mathematical induction, prove each of the following for all n ϵ N: (1 + 3/1) (1 + 5/4)

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Using the principle of mathematical induction, prove each of the following for all n ϵ N:

$(1+\frac{3}1)$$(1+\frac{5}4)$$(1+\frac{7}9)$......$\left\{1+\frac{(2n+1)}{n^2}\right\}$ = (n + 1)2

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To Prove:

$(1+\frac{3}1)$ x $(1+\frac{5}4)$ x $(1+\frac{7}9)$ x...... x $\left\{1+\frac{(2n+1)}{n^2}\right\}$ = (n + 1)2

Let us prove this question by principle of mathematical induction (PMI)

Let P(n): $(1+\frac{3}1)$ x $(1+\frac{5}4)$ x $(1+\frac{7}9)$ x...... x $\left\{1+\frac{(2n+1)}{n^2}\right\}$ = (n + 1)2

For n = 1

LHS = 1 + $\frac{3}1$ = 4

RHS = (1 + 1)= 4

Hence, LHS = RHS

P(n) is true for n = 1

Assume P(k) is true

$(1+\frac{3}1)$ x $(1+\frac{5}4)$ x $(1+\frac{7}9)$ x...... x $\left\{1+\frac{(2k+1)}{k^2}\right\}$ = (k + 1)2....(1)

We will prove that P(k + 1) is true [Now writing the second last term] = RHS

LHS = RHS

Therefore, P (k + 1) is true whenever P(k) is true

By the principle of mathematical induction, P(n) is true for

Where n is a natural number

Hence proved.