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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.

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