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Chapter 12: Problem 57

Expand the partial sum and find its value. \(\sum_{k=1}^{5} k(k+1)\)

Short Answer

Expert verified
The value of the sum is 70.

Step by step solution

01

Identify the Series

Understand the given series is \(\sum_{k=1}^{5} k(k+1)\).
02

Expand the Series

Write down each term by plugging in values of k from 1 to 5: \(1(1+1) + 2(2+1) + 3(3+1) + 4(4+1) + 5(5+1)\).
03

Simplify Each Term

Calculate each multiplication: \(1(2) + 2(3) + 3(4) + 4(5) + 5(6)\) becomes 2 + 6 + 12 + 20 + 30.
04

Sum the Terms

Add the simplified terms together: 2 + 6 + 12 + 20 + 30 = 70.

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Key Concepts

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

partial sum
In the context of a series, a 'partial sum' represents the sum of the first several terms in that series. When you see a summation notation like \(\sum_{k=1}^{5} k(k+1)\), it is asking for the sum of the first five terms of the sequence defined by the expression \(k(k+1)\). Partial sums help us understand the behavior of the series over smaller segments. To find the partial sum, you add up each term's value from the starting index up to the given endpoint. In this exercise, we've identified the terms as \(1(2) + 2(3) + 3(4) + 4(5) + 5(6)\), which we then calculated to get the total sum of 70.
series expansion
Series expansion is the process of expressing the given summation as individual terms. For the series \(\sum_{k=1}^{5} k(k+1)\), we expanded it by calculating each term independently for values of \( k \) from 1 to 5. This is done by substituting \(k\) in the expression \(k(k+1)\)), resulting in the sequence: \(k^2 + k\). By solving for every \(k\) in the range, we can list the terms: \(2, 6, 12, 20, 30\). Expansion helps to both visualize and compute the series more effectively.
summation notation
Summation notation is a shorthand way of writing sums of sequences. In the expression \(\sum_{k=1}^{5} k(k+1)\), \(\sum\) symbolizes summation, \(k=1\) is the starting index, \(5\) is the ending index, and \(k(k+1)\) is the function of \(k\) we are summing. This notation is incredibly useful in mathematics because it concisely conveys the operation of adding multiple terms in a series. Understanding how to translate summation notation into expanded form is essential for solving series-related problems.
algebraic simplification
Algebraic simplification involves reducing expressions into simpler forms to make calculations easier. In the step-by-step solution, after expanding the series \(\sum_{k=1}^{5} k(k+1)\), we simplified the terms by performing the multiplication: \(1(2), 2(3), 3(4), 4(5), 5(6)\). Each term was then simplified to become \(2, 6, 12, 20, 30\) respectively. Simplification makes it straightforward to sum the terms and reach the final answer, which in this case is 70. Mastering algebraic simplification can significantly improve problem-solving efficiency in math.

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