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Chapter 12: Problem 58

Expand the partial sum and find its value. \(\sum_{k=1}^{5} k(2 k-3)\)

Short Answer

Expert verified
The value of the sum is 65.

Step by step solution

01

- Write Out the Sum

The given sum notation is \(\sum_{k=1}^{5} k(2 k-3)\). This means we will evaluate and sum the expression \(k(2k-3)\) for values of \(k\) from 1 to 5.
02

- Apply the Expression

Calculate each term \(k(2k-3)\) for \(k = 1, 2, 3, 4, 5\).For \(k = 1,\ (1)(2(1)-3) = (1)(2-3) = (1)(-1) = -1\).For \(k = 2,\ (2)(2(2)-3) = (2)(4-3) = (2)(1) = 2\).For \(k = 3,\ (3)(2(3)-3) = (3)(6-3) = (3)(3) = 9\).For \(k = 4,\ (4)(2(4)-3) = (4)(8-3) = (4)(5) = 20\).For \(k = 5,\ (5)(2(5)-3) = (5)(10-3) = (5)(7) = 35\).
03

- Sum the Results

Add the values obtained in Step 2: \-1 + 2 + 9 + 20 + 35\This gives: \(-1 + 2 = 1 \) \(+ 9 = 10 \) \(+ 20 = 30 \) \(+ 35 = 65\).

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Key Concepts

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

Summation Notation
Summation notation, denoted by the Greek letter sigma \(\sum\), is a concise way to represent the sum of a sequence of values.
It consists of three parts: the summation symbol \(\sum\), the index of summation (often represented by \(k\)), and an expression involving the index.
The example provided is \(\sum_{k=1}^{5} k(2 k-3)\), which means, start from \(k = 1\) and go up to \(k = 5\), calculating the expression \((2k-3)\) for each value of \(k\).
Finally, add all those resulting values together.
Understanding summation notation transforms complex series into manageable parts, making it easier to perform calculations and spot patterns.
Arithmetic Series
An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant.
In our given exercise, the terms calculated from the expression \((2k-3)\) are \(-1, 2, 9, 20, \) and \(35\).
While not strictly an arithmetic sequence (since the difference between terms is not constant), the process involves summing a series of terms derived from an expression.
So understanding the idea behind arithmetic series helps in identifying the progression of terms and simplifying such sums.
Analyzing the step-by-step results can also reveal arithmetic patterns, aiding in faster computation.
Algebraic Expressions
An algebraic expression is a mathematical phrase combining numbers, variables, and operational symbols.
In the example \(\sum_{k=1}^{5} k(2 k-3)\), the part \(k(2k-3)\) is an algebraic expression.
To solve it, one must substitute the given values of the variable \(k\).
Notice that \(k(2k - 3)\) is simplified using mathematical operations: first, calculate inside the parentheses, then multiply by \(k\).
Practicing algebraic manipulations builds a strong foundation for handling complex mathematical problems.

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Most popular questions from this chapter

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