Calculating the Sum of a Series with Repeating Digits: An SEO-Optimized Guide

When dealing with series in mathematics, one intriguing topic is the sum of sequences with repeating digits. A series 777777... n terms is a classic example that involves both computational complexity and mathematical elegance. In this article, we will explore how to calculate the sum of such a series, present the formula, and apply it to specific cases.

Understanding the Series: 777777... n Terms

To start, let's understand the nature of the series. The series in question is a sequence composed of the digit 7 repeated n times. For example, if n 5, the series would be 77777.

Generalized Formula for the Sum

The key to solving this problem lies in formulating a generalized equation. Each k-th term in the series can be represented as:

( a_k frac{7 times (10^k - 1)}{9} )

Using this representation, the sum of the first n terms (S_n) of the series can be expressed as:

( S_n frac{7}{9} left( sum_{k1}^{n} 10^k - sum_{k1}^{n} 1 right) )

This can be further simplified into:

( S_n frac{7}{81} 10^{n 1} - 10 - 9n )

Deriving the Formula Step-by-Step

Let's break down the steps to derive the formula for the sum of the series 777777... n terms:

Express each term in the series as ( 7 times 10^0 7 times 10^1 7 times 10^2 ldots 7 times 10^{n-1} ) Note that each term can be rewritten as: ( a_k 7 times frac{10^k - 1}{9} ) The sum of the series up to n terms is given by: ( S_n frac{7}{9} left( sum_{k1}^{n} 10^k - n right) ) Using the formula for the sum of a geometric series: ( sum_{k1}^{n} 10^k frac{10^{n 1} - 10}{9} ) Finally, the total sum simplifies to: ( S_n frac{7}{81} 10^{n 1} - 10 - 9n )

Application and Examples

Let's apply this formula to a specific case where n 5:

For n 5, we have 777777. Plugging this into our formula:

( S_5 frac{7}{81} 10^6 - 10 - 9 times 5 )

Calculating this:

( S_5 frac{7000000}{81} - 10 - 45 approx 86407.407 - 10 - 45 )

Thus, the sum is approximately:

( S_5 86415 )

Alternative Approach

Alternatively, we can add the terms manually:

7 77 777 7777 77777

Summing these up:

7 77 777 7777 77777 86415

Conclusion

In conclusion, the sum of the series 777777... n terms can be effectively calculated using a generalized formula derived from the properties of geometric series. This article has provided a detailed explanation and a step-by-step guide to applying the formula to various cases.

For further exploration and practice, consider applying this formula to other series or to larger n values to better understand its practical applications.