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Chapter 7: Problem 16

In Exercises 15-24, evaluate the geometric series. \(1+2+4+\cdots+2^{100}\)

Short Answer

Expert verified
The sum of the given geometric series up to the 100th term is \(2^{100} - 1\).

Step by step solution

01

Identify the given values

The given series is a geometric series with first term a_1 = 1, common ratio r = 2, and number of terms n = 100.
02

Use the geometric series sum formula

We will use the geometric series sum formula S_n = a_1 * (1 - r^n) / (1 - r) to find the sum of the series up to the 100th term.
03

Substitute the given values in the formula

Now, substitute the given values in the formula: S_100 = (1 * (1 - 2^{100})) / (1 - 2)
04

Simplify the expression

We can simplify the expression further: S_100 = (1 - 2^{100}) / (-1)
05

Solve for S_100

Finally, we can solve for S_100: S_100 = 2^{100} - 1 The sum of the given geometric series up to the 100th term is 2^{100} - 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Geometric Series Formula
Geometric series are a fascinating concept in mathematics where each term is derived by multiplying the previous term by a constant known as the common ratio. The formula to find the sum of the first \( n \) terms of a geometric series is:
  • \( S_n = a_1 \times \frac{1 - r^n}{1 - r} \)
Let's break this down:
  • \( S_n \) represents the sum of the series up to the \( n^{th} \) term.
  • \( a_1 \) is the first term of the series.
  • \( r \) is the common ratio.
  • \( n \) is the number of terms you are summing up.
The formula helps us find the sum without having to go through the process of adding each term manually. This is particularly useful for series with a large number of terms. In the exercise, this formula allowed us to calculate the sum of 100 terms swiftly.
Identifying the Common Ratio
The common ratio is a critical part of understanding a geometric series. It determines how each term in the series progresses from the previous one. For the series given in the exercise, the common ratio \( r \) is 2. But how do we know this?The common ratio is found by dividing any term by its preceding term in an ideal geometric series:
  • If as an example we use terms 2 and 1: \( r = \frac{2}{1} = 2 \).
  • Onwards: \( \frac{4}{2} = 2 \), confirming consistency.
The consistent common ratio is what classifies our series as geometric. Knowing the common ratio is essential as it influences the overall behavior of the series and helps determine the correct sum.
Calculating the Sum of the Series
When we talk about the sum of a geometric series, we're interested in finding the added value of all its terms up to a specific point. Using the formula from earlier, the sum \( S_n \) for our series was determined as:
  • \( S_{100} = \frac{1 \times (1 - 2^{100})}{1 - 2} \)
  • Simplified to \( \frac{1 - 2^{100}}{-1} \)
  • Final result: \( 2^{100} - 1 \)
In this expression:
  • The numerator \( 1 - 2^{100} \) represents the growth effect from the geometric progression.
  • The denominator \( 1 - 2 \) signifies adjusting this effect back to the initial scale by the first term.
So, the sum captures how quickly the series grows, driven primarily by that common ratio, and gives us the total up to any chosen term.

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Most popular questions from this chapter

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