Find the sum of two consecutive number for $(13)^{2}$ .

84

and

85

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83

and

84

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86

and

82

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81

and

80

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Solution

The correct option is A $84$ and $85$
Given square $n^{2}$ can be written as sum of two consecutive numbers $\frac{n^{2} - 1}{2}$ and $\frac{n^{2} + 1}{2}$ when $n$ is odd.
Given $n^{2} = (13)^{2} = 169$
$\frac{n^{2} - 1}{2} = \frac{(13)^{2} - 1}{2} = 84$
$\frac{n^{2} + 1}{2} = \frac{(13)^{2} + 1}{2} = 85$
So, the sum of two consecutive numbers $= 84 + 85 = 169.$

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