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Chapter 11: Problem 42

Evaluate each series. $$\sum_{i=3}^{6}\left(2 i^{2}+1\right)$$

Short Answer

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176

Step by step solution

01

- Understand the Summation Notation

The summation notation \(\sum_{i=3}^{6}\left(2 i^{2}+1\right)\) means that you need to evaluate the expression \(2i^{2} + 1\) for each integer value of \(i\) from 3 to 6, and then find the sum of these values.
02

- Evaluate the Expression for Each Integer

Evaluate the expression \(2i^{2} + 1\) for each value of \(i\) from 3 to 6:For \(i = 3\), \(2(3)^{2} + 1 = 2 \times 9 + 1 = 19\)For \(i = 4\), \(2(4)^{2} + 1 = 2 \times 16 + 1 = 33\)For \(i = 5\), \(2(5)^{2} + 1 = 2 \times 25 + 1 = 51\)For \(i = 6\), \(2(6)^{2} + 1 = 2 \times 36 + 1 = 73\)
03

- Sum the Evaluated Values

Add the values obtained from Step 2:\(19 + 33 + 51 + 73\)Calculate the sum: \(19 + 33 = 52\)\(52 + 51 = 103\)\(103 + 73 = 176\)

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Key Concepts

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

series evaluation
Understanding how to evaluate a series is crucial in mathematics. A series is simply the sum of the terms of a sequence. In the given exercise, the series is expressed using summation notation: \(\sum_{i=3}^{6}\left(2 i^{2}+1\right)\). This notation tells us to calculate the expression \(2i^{2} + 1\) for each integer value of \(i\) from 3 to 6, and then sum up these results.
By breaking down the problem as outlined in the step-by-step solution, we can see exactly how the calculations are performed:
  • For \(i = 3\), we get \(2(3)^{2} + 1 = 19\).
  • For \(i = 4\), we get \(2(4)^{2} + 1 = 33\).
  • For \(i = 5\), we get \(2(5)^{2} + 1 = 51\).
  • For \(i = 6\), we get \(2(6)^{2} + 1 = 73\).

Finally, summing these values gives us the total of \(19 + 33 + 51 + 73 = 176\). This step-by-step method ensures clarity and accuracy in evaluating the series.
summation formula
Summation formulas can simplify the process of finding the sum of a series. For instance, the formula for the sum of the first \(n\) squares is given by:
\[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]
However, in our exercise, we are not using such a direct formula. Instead, we evaluate each term individually within a specific range (from \(i = 3\) to \(6\)). Knowing different summation formulas can be very useful:
  • Arithmetic series: \[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \]
  • Geometric series with ratio \(r\): \[ \sum_{k=0}^{n} ar^k = a \frac{1-r^{n+1}}{1-r} \] (for \(r e 1\))
    By applying appropriate summation formulas, you can significantly speed up the process of series evaluation in more complex problems.
integer sequence
An integer sequence is a list of numbers where each number is an integer. In our exercise, the integer sequence is implicitly given by the range of \(i\) values (from 3 to 6). Sequences can take various forms and be defined by different rules:
  • Arithmetic sequence: each term is obtained by adding a constant to the previous term. Example: 2, 4, 6, 8, ...
  • Geometric sequence: each term is found by multiplying the previous term by a fixed non-zero number. Example: 2, 6, 18, 54, ...
  • Square sequence: the terms are the squares of integers. Example: 1, 4, 9, 16, ...

In the context of our exercise, we can notice that we are looking at expressions of the form \(2i^2 + 1\) for successive integers. Understanding these sequences helps in breaking down the problem and computing the overall sum more easily.

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Most popular questions from this chapter

The series $$e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\cdots+\frac{a^{n}}{n !}$$ where \(n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot n,\) can be used to approximate the value of \(e^{a}\) for any real number \(a\). Use the first eight terms of this series to approximate each expression. Compare this approximation with the value obtained on a calculator. (a) \(e\) (b) \(e^{-1}\)

Write the terms of \(\sum_{i=1}^{4} f\left(x_{i}\right) \Delta x,\) with \(x_{1}=0, x_{2}=2, x_{3}=4, x_{4}=6,\) and \(\Delta x=0.5,\) for =ach function. Evaluate the sum. $$f(x)=\frac{5}{2 x-1}$$

Use summation notation to write each series. $$1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots-\frac{1}{128}$$

One of the most famous sequences in mathematics is the Fibonacci sequence, $$1,1,2,3,5,8,13,21,34,55, \dots$$ Male honeybees hatch from eggs that have not been fertilized, so a male bee has only one parent, a female. On the other hand, female honeybees hatch from fertilized eggs, so a female has two parents, one male and one female. The number of ancestors in consecutive generations of bees follows the Fibonacci sequence. Draw a tree showing the number of ancestors of a male bee in each generation following the description given above.

How many different samples of 4 apples can be drawn from a crate of 25 apples?
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