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Question Number 137588 by bramlexs22 last updated on 04/Apr/21

F o r a p o s i t i v e n u m b e r n, l e t

f (n) b e t h e v a l u e o f

f (n) = \frac{4 n + \sqrt{4 n^{2} - 1}}{\sqrt{2 n + 1} + \sqrt{2 n - 1}}

c a l c u l a t e f (1) + f (2) + f (3) + \dots + f (40) .

Answered by bemath last updated on 04/Apr/21

\Leftrightarrow f (n) = \frac{{(\sqrt{2 n + 1})}^{2} + {(\sqrt{2 n - 1})}^{2} + \sqrt{(2 n + 1) (2 n - 1)}}{\sqrt{2 n + 1} + \sqrt{2 n - 1}}

f (n) = \frac{{(\sqrt{2 n + 1})}^{3} - {(\sqrt{2 n - 1})}^{3}}{2}

f (1) + f (2) + f (3) + \dots + f (40)

= \frac{\sqrt{3^{3}} - \sqrt{1^{3}}}{2} + \frac{\sqrt{5^{3}} - \sqrt{3^{3}}}{2} + \frac{\sqrt{7^{3}} - \sqrt{5^{3}}}{2} + \dots + \frac{\sqrt{81^{3}} - \sqrt{79^{3}}}{2}

[t e l e s c o p y s e r i e s]

f (1) + f (2) + f (3) + \dots + f (40)

= \frac{\sqrt{81^{3}} - \sqrt{1^{3}}}{2} = \frac{9^{3} - 1}{2} = \frac{728}{2} = 364

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