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find the sum of each of the following sequences up to n terms.
(a) 8, 88, 888, . . .
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So, we want to find the sum
R = 8 + 88 + 888 + . . . + 888...8, (1)
where the last number is written using "n" digits of 8.
Consider another sum
T = = 9 + 99 + 999 + . . . + 999...9, (2)
where the last number is written using "n" digits of 9.
Add n to T. It is the same as to add "1" to each term of the sum (2). Therefore,
T + n = 10 + 100 + 1000 + . . . + 10^n. (3)
The sum (3) is the sum of the first n terms of the geometric progression with the first term 10 and the common ratio of 10.
So, the sum (3) is equal to
T + n = = .
Thus you have
T = , or
= , which implies
R = .
It is the final formula and the ANSWER : R = 8 + 88 + 888 + . . . + 888...8 = .
Check. Let us take n = 2. Then from one side, R = 8 + 88 = 96.
From the other side, R = = = = = = 88. ! Correct !
Let us take n = 3. Then from one side, R = 8 + 88 + 888 = 984.
From the other side, R = = = = = = 984. ! Correct !