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Finding Square of a Number Using Identity

Evaluate : i ...

Question

Evaluate :

(i) $37^{2} - 36^{2}$ (ii) $85^{2} - 84^{2}$

(iii) $101^{2} - 100^{2}$

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Solution

(i) $37^{2} - 36^{2}$

Using property, for any natural number n,

$(n + 1)^{2} - n^{2} = (n + 1) + n \Rightarrow (36 + 1)^{2} - 36^{2} = (36 + 1) + 36 \Rightarrow 37^{2} - 36^{2} = 37 + 36 \Rightarrow 37^{2} - 36^{2} = 73$

(ii) $85^{2} - 84^{2}$

Using property, for any natural number n,

$(n + 1)^{2} - n^{2} = (n + 1) + n \Rightarrow (84 + 1)^{2} - 84^{2} = (84 + 1) + 84 \Rightarrow 85^{2} - 84^{2} = 85 + 84 \Rightarrow 85^{2} - 84^{2} = 169$

(iii) $101^{2} - 100^{2}$

Using property, for any natural number n,

$(n + 1)^{2} - n^{2} = (n + 1) + n \Rightarrow (100 + 1)^{2} - 100^{2} = (100 + 1) + 100 \Rightarrow 101^{2} - 100^{2} = 101 + 100 \Rightarrow 101^{2} - 100^{2} = 201$

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