Solve 1-n-1-1-n-2-18-9n-13

(1/n+1)+(1/n+2)=(18/9x+13) (n+2+n+1/(n+1)(n+2))=(18/9n+13) (2n+3)(9n+13)=18(n+1)(n+2) 18^n2+26n+27n+39=18(n^2+3n+2) 18n^2+53n+39=18n^2+54n+ ...

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Solving Equations with Variables on Both Sides

Solve 1/n+1...

Question

Solve $\frac{1}{n + 1} + \frac{1}{n + 2} = \frac{18}{9 n + 13}$

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Solution

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\frac{1^{3}}{1} + \frac{1^{3} + 2^{3}}{1 + 3} + \frac{1^{3} + 2^{3} + 3^{3}}{1 + 3 + 5} + . . . . n

= \frac{n}{24} (2 n^{2} + 9 n + 13)

S_{n} = \sum_{r = 1}^{n} t r = \frac{1}{6} n (2 n^{2} + 9 n + 13)

\sum_{r = 1}^{n} \sqrt{t r}

13 \sum n = 1 (t^{n} - t^{n + 1}) =

n

\frac{9^{n + 1} + 4^{n + 1}}{9^{n} + 4^{n}} = 6

\frac{9^{n + 1} \times {(3 \frac{- n}{2})}^{- 2} - 27^{n}}{{(3^{m} \times 2)}^{3}} = \frac{1}{729}

m - n = 2

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