$2^{2} + 22^{2} + 222^{2} + . . . . . .$ upto n terms.

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Solution

$S_{n} = 2^{2} + 22^{2} + 222^{2} + . . . . . + n$

$= 2^{2} [1 + 11 + 111 + . . . . . . + n]$

$= \frac{2^{2}}{9} [9 + 99 + 999 + . . . . . + n]$

$= \frac{2^{2}}{9} [(10 - 1) + (10^{2} - 1) + (10^{3} - 1) + . . . . . . . + n]$

$= \frac{2^{2}}{9} {(10 + 10^{2} + . . . . + n) + (1 + 1 + 1..... + n)}$

$= \frac{2^{2}}{9} [\frac{10^{n} - 1}{10 - 1} - n]$

$= \frac{4}{9} [\frac{10^{n} - 9 n - 1}{9}]$

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