Sum of a Series with an Exponential Term: Exploring the Term 32^n - 4

Introduction to Exponential Series

When dealing with series that involve exponential terms, it becomes crucial to understand the nature and sum of these terms. This article focuses on a specific series where each term is defined as Tn 32^n - 4. This series is examined from n 1 to n 100, providing a comprehensive understanding of how to sum such series and the techniques involved.

Understanding the Series

Let's delve into a series defined by the term Tn 32^n - 4. This series is a non-trivial example because it involves both an exponential component and a constant subtraction. The exponential term, 32^n, represents a geometric progression (GP) with a base of 32. The constant subtraction of 4 affects the sum of each term but complicates the overall summation process.

Summing the Series (n 1 to 100)

The objective is to find the sum of the first 100 terms of the series, denoted as S. To achieve this, we can break the problem into simpler sub-problems by separating the series into two components: a geometric series and a constant series.

The Given Series

Let's assume the series to be summed is:

S Σn1100 (32^n - 4)

This can be rewritten as:

S 3Σn1100 2^n - Σn1100 4

We can further break this into two sums:

S S1 - S2

Sum of the Geometric Series (S1)

The first sum, S1, is the sum of the series:

S1 3Σn1100 2^n

This is a geometric series with a common ratio, r 2, and the first term, a 2. The sum of the first n terms of a geometric series is given by:

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

Substituting a 2, r 2, and n 100:

S1 3(2 * (2^100 - 1) / (2 - 1))

S1 3(2^101 - 2)

S1 6 * 2^101 - 6

Sum of the Constant Series (S2)

The second sum, S2, is the sum of 100 instances of the constant 4:

S2 4 * 100

S2 400

Combining the Sums

Substituting the values of S1 and S2 into the equation for S, we get:

S S1 - S2

S (6 * 2^101 - 6) - 400

S 6 * 2^101 - 406

Conclusion

Thus, the sum of the first 100 terms of the series Tn 32^n - 4 is:

S 6 * 2^101 - 406

This demonstrates the application of geometric series properties and the simplification of complex series sums using algebraic manipulation. Understanding such techniques is crucial for advancing in fields such as mathematics, computer science, and data analysis.