Find the sum to n terms of the sequence, 8, 88, 888, 8888…

To find the sum of the sequence 8, 88, 888, 8888..., we need to determine the pattern and then use the formula for the sum of an arithmetic series.

Step 1: Observe the pattern Looking at the sequence, we can see that each term is formed by repeating the digit 8 a certain number of times. The first term is 8, the second term is 88 (8 repeated twice), the third term is 888 (8 repeated three times), and so on.

Step 2: Determine the general term From the pattern, we can deduce that the nth term of the sequence is formed by repeating the digit 8 n times. So, the general term of the sequence can be written as 8 × (10^n - 1)/9.

Step 3: Find the sum of the sequence To find the sum of the sequence, we need to find the sum of the first n terms. The formula for the sum of an arithmetic series is given by S = n/2 × (a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.

In our case, the first term (a) is 8 and the last term (l) is 8 × (10^n - 1)/9. So, the sum of the sequence can be calculated as S = n/2 × (8 + 8 × (10^n - 1)/9).

Step 4: Simplify the formula To simplify the formula, we can multiply both sides by 9 to get rid of the fraction. This gives us 9S = n × (8 + 8 × (10^n - 1)).

Expanding the equation further, we have 9S = n × (8 + 8 × 10^n - 8).

Simplifying, we get 9S = n × 8 × 10^n.

Finally, dividing both sides by 9, we obtain S = n × 8 × 10^n / 9.

So, the sum to n terms of the sequence 8, 88, 888, 8888... is given by the formula S = n × 8 × 10^n / 9.