Question

Find the sum to n terms of the series, 7 + 77 + 777 + ...... .

Answer 1

S_n = 7 + 77 + 777 + ..... to n terms 

= 7 [1 + 11 + 111 + ..... to n terms]

= \(\frac{7}{9}\) [9 + 99 + 999 + ..... to n terms]

= \(\frac{7}{9}\) [(10 – 1) + (100 – 1) + (1000 – 1) + ..... to n terms]

= \(\frac{7}{9}\) [{10 + 10² + 10³ + ..... to n terms} – {1 + 1 + 1 + ..... to n terms}]

= \(\frac{7}{9}\) \(\bigg[\frac{10(10^n-1)}{10-1}-n\bigg]\) = \(\frac{7(10^{n+1})-10}{81}-\frac{7}{9}n.\)