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Chapter 11: Problem 57

Write out and evaluate each sum. $$ \sum_{k=0}^{4} 10^{k} $$

Short Answer

Expert verified
11111

Step by step solution

01

Understand the Sigma Notation

The notation \(\sum_{k=0}^{4} 10^{k}\) represents the summation of the expression \(10^{k}\) for each integer value of \(k\) from 0 to 4.
02

List the Terms of the Sum

Evaluate \(10^{k}\) for each \(k\) from 0 to 4:\(10^{0}, 10^{1}, 10^{2}, 10^{3}, 10^{4}\). This results in the terms: 1, 10, 100, 1000, 10000.
03

Evaluate Each Term

Compute each term: \(10^{0} = 1\) \(10^{1} = 10\) \(10^{2} = 100\) \(10^{3} = 1000\) \(10^{4} = 10000\).
04

Add the Terms Together

Add the evaluated terms: \(1 + 10 + 100 + 1000 + 10000 = 11111\).

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Key Concepts

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

sigma notation
Sigma notation, represented by the symbol \(\sum\), is a concise way of writing the sum of a sequence of numbers. When you see \(\sum_{k=a}^{b} f(k)\), it means you need to sum the function \(f(k)\) starting from \(k=a\) and going up to \(k=b\). In the exercise provided, \(\sum_{k=0}^{4} 10^{k}\), we are summing the powers of 10 from 10^0 to 10^4.
This notation helps to simplify and organize expressions that involve adding many terms, particularly when they follow a regular pattern.
powers of ten
Understanding powers of ten is crucial when evaluating summations that include exponential terms. The expression \(10^{k}\) means 10 raised to the power of \(k\).
Here's a breakdown of powers of ten:
  • 10^0 = 1
  • 10^1 = 10
  • 10^2 = 100
  • 10^3 = 1000
  • 10^4 = 10000
In our sum from the exercise, we calculate each of these powers and then add them together. Recognizing these patterns can help you quickly identify the terms in sequences involving powers of ten.
evaluating sums
Evaluating sums involves calculating and adding the terms given in the sequence. Let鈥檚 go step by step:
  1. Identify the terms to be added as guided by the summation notation.
  2. Calculate each term individually from the sequence. For the sum \(\sum_{k=0}^{4} 10^{k}\), we compute 1, 10, 100, 1000, and 10000.
  3. Add up all these terms: 1 + 10 + 100 + 1000 + 10000.
  4. Combine the results to get the final sum, which in this case, is 11111.
Following these steps systematically ensures you evaluate the sum accurately every time.

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