Find the sum of the sequence 7, 77, 777, 7777, ... to n terms.

Question

Find the sum of the sequence 7, 77, 777, 7777, ... to $n$ terms.

Smart Achievers · Accepted Answer

(1)

$This is not a G.P., however, we can relate it to a G.P. by writing the terms as$

$S_{n} = 7 + 77 + 777 + 7777 + \dots to n terms$

$= \frac{7}{9} [9 + 99 + 999 + 9999 + \dots to n terms]$

$= \frac{7}{9} [(10 - 1) + (10^{2} - 1) + (10^{3} - 1) + (10^{4} - 1) + \dots n terms]$

$= \frac{7}{9} [(10 + 10^{2} + 10^{3} + \dots n terms) - (1 + 1 + 1 + \dots n terms)]$

$= \frac{7}{9} [\frac{10 (10^{n} - 1)}{10 - 1} - n] = \frac{7}{9} [\frac{10 (10^{n} - 1)}{9} - n]$

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