Question

Find the sum of the following to n terms: 7 + 77 + 777 + 7777 + ..........

[Hint: $$S_n = 7(1 + 11 + 111 + 1111 +\ .........)$$ $$= \frac{7}{9} (9 + 99 + 999 + 9999 +\ .........)$$
$$= \frac{7}{9} \left[(10 - 1) + (100 - 1) + (1000 - 1)\ .........\right]$$
$$= \frac{7}{9} (10 + 100 + 1000 +\ ....... -1 -1 -1\ ........\ n \text{ times})$$]

Answer 1

Using the hint:

$$S_n = \frac{7}{9} (10 + 100 + 1000 + \ldots \text{ to } n \text{ terms} - n)$$

Now, $$10 + 100 + 1000 + \ldots + 10^n$$ (n terms)

$$= 10\left(\dfrac{10^n - 1}{10 - 1}\right)$$

$$= \dfrac{10(10^n - 1)}{9}$$

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

$$= \frac{7}{81}\left[10(10^n - 1) - 9n\right]$$

$$= \frac{7(10^{n+1} - 10 - 9n)}{81}$$

Hence, $$S_n = \frac{7}{9}\left[\frac{10(10^n - 1)}{9} - n\right] = \frac{7(10^{n+1} - 10 - 9n)}{81}$$.