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S = 1 + $\frac{4}{7}$ + $\frac{9}{7^{2}}$ + $\frac{16}{7^{3}}$ + $\frac{25}{7^{4}}$ +.................... (A)

$\frac{S}{7}$= $\frac{1}{7}$ + $\frac{4}{7^{2}}$ + $\frac{9}{7^{3}}$ + $\frac{16}{7^{4}}$ +.................... (B)

Substract equation (B) from (A),

$\frac{6S}{7}$ = 1 + $\frac{3}{7}$ + $\frac{5}{7^{2}}$ + $\frac{7}{7^{3}}$ + $\frac{9}{7^{4}}$ + ...................... (C)

$\frac{6S}{7^{2}}$= $\frac{1}{7}$ + $\frac{3}{7^{2}}$ + $\frac{5}{7^{3}}$ + $\frac{7}{7^{4}}$ + ...................... (D)

Substract equation (D) from (C),

$\frac{36S}{49}$ = 1 + $\frac{2}{7}$ + $\frac{2}{7^{2}}$ + $\frac{2}{7^{3}}$ + $\frac{2}{7^{4}}$ + .................

$\frac{36S}{49}$ = 1 + $\frac{2 }{7}$ * $\frac{7 }{6}$

$\frac{36S}{49}$ = $\frac{4}{3}$

S = $\frac{4 * 49}{3* 36}$

S = $\frac{49}{27}$

**Ans- C.**