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Chapter 5: Problem 4

Find the value of the indicated sum. $$ \sum_{l=3}^{8}(l+1)^{2} $$

Short Answer

Expert verified
The sum is 271.

Step by step solution

01

Understand the Summation Notation

The expression \( \sum_{l=3}^{8}(l+1)^{2} \) tells us to evaluate the sum of \( (l+1)^2 \) as \( l \) ranges from 3 to 8. This means we need to calculate \( (4)^2, (5)^2, (6)^2, (7)^2, (8)^2, \) and \( (9)^2 \), and then sum these results.
02

Calculate Each Term

Evaluate each term:- For \( l = 3 \), \( (l+1)^2 = (3+1)^2 = 4^2 = 16 \)- For \( l = 4 \), \( (l+1)^2 = (4+1)^2 = 5^2 = 25 \)- For \( l = 5 \), \( (l+1)^2 = (5+1)^2 = 6^2 = 36 \)- For \( l = 6 \), \( (l+1)^2 = (6+1)^2 = 7^2 = 49 \)- For \( l = 7 \), \( (l+1)^2 = (7+1)^2 = 8^2 = 64 \)- For \( l = 8 \), \( (l+1)^2 = (8+1)^2 = 9^2 = 81 \)
03

Sum the Calculated Terms

Add the results from each calculation:\( 16 + 25 + 36 + 49 + 64 + 81 \).Evaluate:\( 16 + 25 = 41 \) \( 41 + 36 = 77 \) \( 77 + 49 = 126 \) \( 126 + 64 = 190 \) \( 190 + 81 = 271 \)
04

Write the Final Answer

The sum of the series from \( l=3 \) to \( l=8 \) for \( (l+1)^2 \) is \( 271 \).

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Key Concepts

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

series evaluation
Evaluating a series involves finding the total sum of a sequence of numbers. In our exercise, evaluating the series means summing the results of \( (l+1)^2 \) as the variable \( l \) changes from 3 to 8. Series evaluation is like a math treasure hunt, where each number gives us a hint until we find the treasure, which is the final sum.
  • Step 1: Identify the range, here it's from \( l=3 \) to \( l=8 \).
  • Step 2: Calculate the expression \( (l+1)^2 \) for each value of \( l \) in the range.
  • Step 3: Add all calculated results to find the series sum.
Imagine evaluating a series as climbing a flight of stairs with each step representing a new calculated term. Reaching the top gives you the entire summation value.
summation notation
Summation notation is a compact way to illustrated adding up a series of numbers. It is usually represented by the Greek letter sigma (\( \sum \)). In our exercise, \( \sum_{l=3}^{8}(l+1)^{2} \) tells us to sum certain terms from 3 to 8 within the given expression. Summation notation is a staple in calculus for organizing computations neatly and efficiently.
  • Notice the limits of the sum: here, from 3 to 8.
  • Understand each term, \( (l+1)^2 \), in the expression.
  • Follow the summation direction from lower index to upper index.
By using summation notation, mathematicians can simplify and solve complex problems more easily. It conveys a lot of information in a small space, helping us keep everything in order.
mathematical series
A mathematical series is essentially the sum of the terms of a sequence. It鈥檚 like connecting the dots between numbers to complete a picture. In this exercise, we're building a series of square numbers with \( (l+1)^2 \) and adding them up.
  • A sequence is a list of numbers generated by a rule.
  • A series takes that sequence and finds the total sum of it.
  • The terms in our series are \( (4)^2, (5)^2, (6)^2, (7)^2, (8)^2, \) and \( (9)^2 \).
Think of a series as assembling different pieces. Each number is part of a larger story that, when summed, reveals the overall picture or, in our case, the total value of \( 271 \).
algebraic summation
Algebraic summation is the art of adding up numbers using algebra. It's not just about getting numbers together; it's about doing it elegantly using algebraic expressions and techniques. In our exercise, we're using the expression \( (l+1)^2 \) to derive each term algebraically for the given range.
  • Recognize the pattern or rule given by \( (l+1)^2 \).
  • Calculate each term systematically.
  • Add all terms to obtain the final sum.
Algebraic summation is like weaving a tapestry where each thread (term) contributes to the complete pattern (sum). This technique is fundamental in understanding growth patterns, financial calculations, and many other real-world applications.

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Most popular questions from this chapter

. Water leaks out of a 200-gallon storage tank (initially full) at the rate \(V^{\prime}(t)=20-t\), where \(t\) is measured in hours and \(V\) in gallons. How much water leaked out between 10 and 20 hours? How long will it take the tank to drain completely?

Sketch the graph of the given function over the interval \([a, b] ;\) then divide \([a, b]\) into \(n\) equal subintervals. Finally, calculate the area of the corresponding circumscribed polygon. $$ f(x)=x+1 ; a=-1, b=2, n=3 $$

Assuming that \(u\) and \(v\) can be integrated over the interval \([a, b]\) and that the average values over the interval are denoted by \(\bar{u}\) and \(\bar{v}\), prove or disprove that (a) \(\bar{u}+\bar{v}=\overline{u+v}\) (b) \(k \bar{u}=\overline{k u}\), where \(k\) is any constant; (c) if \(u \leq v\) then \(\bar{u} \leq \bar{v}\).

Decide whether the given statement is true or false. Then justify your answer. If \(\int_{a}^{b} f(x) d x=0\), then \(f(x)=0\) for all \(x\) in \([a, b]\).

first recognize the given limit as a definite integral and then evaluate that integral by the Second Fundamental Theorem of Calculus. $$ \lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left[\sin \left(\frac{\pi i}{n}\right)\right] \frac{\pi}{n} $$
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