Question

$(\frac{n}{2} ) + ( \frac{n + 1}{2} ) = n { }^{2}$
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Answer 1

Answer:

$\frac{n}{2} + \frac{n + 1}{2} = {n}^{2} \\ \frac{n + n + 1}{2} = {n}^{2} \\ \frac{2n + 1}{2} = {n}^{2} \\ 2n + 1 = {n}^{2} \times 2 \\ 2n + 1 = 2 {n}^{2} \\ 2 {n}^{2} - 2n - 1 = 0 \\ n = \frac{ - b± \sqrt{{b}^{2} - 4ac}}{2a} \\ n = \frac{ - ( - 2)± \sqrt{ {( - 2)}^{2} - 4 \times 2 \times ( - 1) } }{2 \times 2} \\ n = \frac{2± \sqrt{4 - ( - 8)} }{4} \\ n = \frac{2± \sqrt{4 + 8} }{4} \\ n = \frac{2± \sqrt{12} }{4} \\ n = \frac{2± \sqrt{4 \times 3 } }{4} \\ n = \frac{2± \sqrt{ {2}^{2} \times 3 } }{4} \\ n = \frac{2±2 \sqrt{3} }{4} \\ n = \frac{2 + 2 \sqrt{3} }{4} \: \: \: or \: \: \: n = \frac{2 - 2 \sqrt{3} }{4} \\ \boxed{ \large n = \underline{ \underline{ \frac{1 + \sqrt{3} }{2}}} \: \: \: or \: \: \: n = \underline{ \underline{ \frac{1 - \sqrt{3} }{2} }}}$