Find the sum of n terms of the series5+55+555+.......................

To find the sum of the first n terms of the series \(5 + 55 + 555 + \ldots\), we can start by identifying a pattern in the terms of the series. Each term can be expressed in a more manageable mathematical form, which allows us to derive a formula for the sum.

Understanding the Series

The series consists of terms that are formed by the digit '5' repeated in increasing quantities. Let's break down the first few terms:

• First term: 5
• Second term: 55 = 5 × 11
• Third term: 555 = 5 × 111

Notice that each term can be represented as follows:

• First term: 5
• Second term: 5 × (10 + 1) = 5 × 11
• Third term: 5 × (100 + 10 + 1) = 5 × 111

General Term Representation

From this, we can see that the k-th term of the series can be expressed as:

a_k = 5 × (10^{k-1} + 10^{k-2} + ... + 10^0)

This is a geometric series where the first term is 1 and the common ratio is 10. The sum of this geometric series can be calculated using the formula:

S = a \frac{(r^n - 1)}{(r - 1)}

In our case, for k terms, the sum becomes:

a_k = 5 × \frac{(10^k - 1)}{9}

Summing the First n Terms

Now, if we want the sum of the first n terms, we need to compute:

S_n = a_1 + a_2 + a_3 + ... + a_n

Substituting our expression for each term, we get:

S_n = 5 × \left(\frac{(10^1 - 1)}{9} + \frac{(10^2 - 1)}{9} + ... + \frac{(10^n - 1)}{9}\right)

Factoring out the common terms, this simplifies to:

S_n = \frac{5}{9} \left((10^1 + 10^2 + ... + 10^n) - n\right)

Final Calculation

The sum of the powers of ten can be calculated using the geometric series formula again:

10^1 + 10^2 + ... + 10^n = 10 \frac{(10^n - 1)}{(10 - 1)} = \frac{10(10^n - 1)}{9}

Substituting this back into our equation gives:

S_n = \frac{5}{9} \left(\frac{10(10^n - 1)}{9} - n\right)

Simplifying this will yield the final expression for the sum of the first n terms of the series.

Conclusion

Thus, the sum of the first n terms of the series \(5 + 55 + 555 + \ldots\) can be expressed as:

S_n = \frac{5}{81}(10^{n+1} - 10 - 9n)

This formula allows you to easily compute the sum for any positive integer n. Simply plug in the value of n, and you’ll have your answer!