Question

What is the sum of `7+77+777+7777+...` to `n` terms ?

Answer 1

`sum_(k=1)^n a_k = 70/81(10^n-1) - 7/9n`

Explanation:

Note that `7/9 = 0.bar(7)`

Hence we can write a formula for the `k`th term:

`a_k = 7/9(10^k - 1)" "` for `k = 1,2,3...`

Let:

`b_k = 7/9(10^k) = 70/9*10^(k-1)`

This is in the form `b_k = b*r^(k-1)`, the general term of a geometric series with initial term `b = 70/9` and common ratio `r = 10`

The sum to `n` terms is given by the formula:

`S_n = (b(r^n-1))/(r-1) = 70/9*(10^n-1)/(10-1) = 70/81*(10^n-1)`

Hence:

`sum_(k=1)^n a_k = 70/81(10^n-1) - 7/9n`