Solve- 25 - 352 - 253 - 354 - 255 - 356 - -

S = 25 + 35^2 + 25^3 + 35^4 + 25^5 + 35^6 + ... = (25 + 25^3 + 25^5 + ...) + (35^2 + 35^4 + 35^6 + ...) = 25[1 + 15^2 + 15^4 + ...] + 35^2[1 + 15 ...

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Property 7

Solve: 25 +...

Question

Solve: $\frac{2}{5} + \frac{3}{5^{2}} + \frac{2}{5^{3}} + \frac{3}{5^{4}} + \frac{2}{5^{5}} + \frac{3}{5^{6}} + . . . . . .$

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(\frac{1}{5} + \frac{2}{5^{2}} + \frac{3}{5^{3}}) + (\frac{1}{5^{4}} + \frac{2}{5^{5}} + \frac{3}{5^{6}}) + . . . . . . . .

7

Find the sum of the following series to infinity :

(i) $1 - \frac{1}{3} + \frac{1}{3^{2}} - \frac{1}{3^{3}} + \frac{1}{3^{4}} + . . . . \infty$

(ii) $8 + 4 \sqrt{2} + 4 + . . . . \infty$

(iii) $\frac{2}{5} + \frac{3}{5^{2}} + \frac{2}{5^{3}} + \frac{3}{5^{4}} + . . . . . \infty$ .

(iv) 10 - 9 + 8.1 - 7.29 + ...... $\infty$

(v) $\frac{1}{3} + \frac{1}{5^{2}} + \frac{1}{3^{3}} + \frac{1}{5^{4}} + \frac{1}{3^{5}} + \frac{1}{5^{6}} + . . . . . \infty$

(2^{51} + 2^{52} + 2^{53} + 2^{54} + 2^{55})

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{3} \frac{1}{3} \frac{7}{3} \end{matrix} \begin{matrix} 1 \frac{2}{3} 2 \end{matrix} \begin{matrix} \frac{5}{3} \frac{4}{3} \frac{2}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{5} \frac{1}{5} \frac{7}{5} \end{matrix} \begin{matrix} \frac{3}{5} \frac{2}{5} \frac{6}{5} \end{matrix} \begin{matrix} 1 \frac{4}{5} \frac{2}{5} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

3 A - 5 B

\frac{2^{- 3} \times 5^{- 3} \times 10^{2} \times 25}{5^{4} \times 2^{- 5}}

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