find the sum of the nth term:- 5 + 55 + 555......................

The given series is a special type of series known as a geometric series. In a geometric series, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

However, this series doesn't seem to follow the typical geometric series pattern. But if we look closely, we can see that each term is actually 5 times a number made up of n '1's. For example, the first term is 51, the second term is 511, the third term is 5*111, and so on.

So, we can rewrite the series as:

51 + 511 + 5111 + 51111 + ... + 5*111...n times

Now, we can factor out the 5:

5 * (1 + 11 + 111 + 1111 + ... + 111...n times)

Now, the series inside the parentheses is a geometric series with a first term (a) of 1 and a common ratio (r) of 10.

The sum (S) of the first n terms of a geometric series can be found using the formula:

S = a * (1 - r^n) / (1 - r)

Substituting the values we have:

S = 1 * (1 - 10^n) / (1 - 10)

S = (1 - 10^n) / -9

Now, don't forget we factored out a 5 at the beginning. So, the sum of the first n terms of the original series is:

5 * S = 5 * [(1 - 10^n) / -9]

Simplify to get the final answer:

Sum = -5/9 * (1 - 10^n)