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Chapter 13: Problem 52

Expand each sum. \(\sum_{k=1}^{n}(k+1)^{2}\)

Short Answer

Expert verified
\( 4 + 9 + 16 + \ldots + (n+1)^2 \)

Step by step solution

01

- Understand the Sum Notation

The given expression \(\sum_{k=1}^{k=n}(k+1)^{2}\) is a summation which involves adding up all the terms \( (k+1)^2 \) for values of \( k \) ranging from 1 to \( n \).
02

- Calculate the First Few Terms

Calculate the first few terms of the sum to understand the pattern:For \( k=1 \): \ ((1+1)^2 = 2^2 = 4) \.For \( k=2 \): \ ((2+1)^2 = 3^2 = 9) \.For \( k=3 \): \ ((3+1)^2 = 4^2 = 16) \.So the first few terms are 4, 9, 16, etc.
03

- Generalize the Sum

The general term for each value of \( k \) is \( (k+1)^2 \). To expand this sum, you write out each of these terms from 1 to \( n \). The expanded form will then be \ 4 + 9 + 16 + \ldots\ + (n+1)^2 \.

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Key Concepts

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

Sum Notation
Sum notation, also known as summation or sigma notation, is a concise way of representing the addition of a sequence of numbers. In our case, we use the Greek letter sigma (\( \sum \)) to denote the sum. The expression \( \sum_{k=1}^{n}(k+1)^{2} \) tells us to add up the terms \( (k+1)^2 \) for values of \( k \) from 1 to \( n \). So, we need to evaluate each term inside the summation by replacing \( k \) with every integer from 1 to \( n \). Sum notation helps simplify complex summations and makes it easier to understand what needs to be calculated.
Expansion of Series
To expand a series, you write out all the terms that you would add together as per the sum notation. Let's take a closer look using our example. The sum \( \sum_{k=1}^{n}(k+1)^{2} \) means we'll evaluate the term \( (k+1)^2 \) for each \( k \) from 1 up to \( n \). Here are the first few terms expanded:
For \( k=1 \): \( (1+1)^2 = 2^2 = 4 \)
For \( k=2 \): \( (2+1)^2 = 3^2 = 9 \)
For \( k=3 \): \( (3+1)^2 = 4^2 = 16 \)
So, if we continue this process up to \( n \), the expanded form looks like:
\( 4 + 9 + 16 + \ldots + (n+1)^2 \). This way, we can see each individual term rather than just the compact sum notation.
Pattern Recognition
Recognizing patterns in the series is key to understanding and solving summation problems. In our example, once we start expanding the series, we notice the terms follow a quadratic pattern: 4, 9, 16, etc. Each term is a square of consecutive integers starting from 2:
  • 4 is \( 2^2 \)
  • 9 is \( 3^2 \)
  • 16 is \( 4^2 \)
Identifying this pattern helps us write out and understand the general term for the series. The more terms you calculate, the easier it becomes to spot these patterns, aiding in quicker and more accurate solutions for larger values of \( n \).
General Term
The general term of a series lets us describe any term in the sequence based on its position. For the summation \( \sum_{k=1}^{n}(k+1)^{2} \), the general term for any given \( k \) is \( (k+1)^2 \). This means, for any \( k \), you simply add 1 to \( k \), then square the result.
It's crucial to understand how to find and use the general term because it allows us to expand, simplify, and sometimes even find closed-form solutions. In our problem, identifying the general term \( (k+1)^2 \) helps quickly identify the exact value of each term in the sequence, making the expansion process straightforward and accurate.

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Most popular questions from this chapter

Suppose you were offered a job in which you would work 8 hours per day for 5 workdays per week for 1 month at hard manual labor. Your pay the first day would be 1 penny. On the second day your pay would be two pennies; the third day 4 pennies. Your pay would double on each successive workday. There are 22 workdays in the month. There will be no sick days. If you miss a day of work, there is no pay or pay increase. How much do you get paid if you work all 22 days? How much do you get paid for the 22nd workday? What risks do you run if you take this job offer? Would you take the job?

True or False A function is a relation between two sets \(D\) and \(R\) so that each element \(x\) in the first set \(D\) is related to exactly one element \(y\) in the second set \(R\)

Suppose that, throughout the U.S. economy, individuals spend \(90 \%\) of every additional dollar that they earn. Economists would say that an individual's marginal propensity to consume is \(0.90 .\) For example, if Jane earns an additional dollar, she will spend \(0.9(1)=\$ 0.90\) of it. The individual who earns \(\$ 0.90\) (from Jane) will spend \(90 \%\) of it, or \(\$ 0.81 .\) This process of spending continues and results in an infinite geometric series as follows: $$1,0.90,0.90^{2}, 0.90^{3}, 0.90^{4}, \ldots$$ The sum of this infinite geometric series is called the multiplier. What is the multiplier if individuals spend \(90 \%\) of every additional dollar that they earn?

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ 2,4,6,8, \ldots $$

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 4\left(-\frac{1}{2}\right)^{k-1} $$
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