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Precalculus : Sequences and Series

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Example Question #31 : Sequences And Series

What is the 10th term in the series:

1, 5, 9, 13, 17....

Possible Answers:

Correct answer:

Explanation:

The pattern in this arithmetic series is that each term is created by adding 4 to the previous one. You can then continue the series by continuing to add 4s until you've gotten to the tenth term:

1, 5, 9, 13, 17, 21, 25, 29, 33, 37

The correct answer, then, is 37.

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Example Question #1 : Partial Sums Of Series

For the sequence

a_1=1

a_2=1

a_3=2

a_4=3

a_5=5

a_6=8

$\cdot$

Determine $\sum_{i=1}^{6}a_i$ .

Possible Answers:

Correct answer:

Explanation:

$\sum_{i=1}^{6}a_i$ is defined as the sum of the terms a_i from i=1 to i=6

Therefore, to get the solution we must add all the entries from a_i from i=1 to i=6 as follows.

a_1+a_2+a_3+a_4+a_5+a_6=1+1+2+3+5+8=20

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Example Question #1 : Partial Sums Of Series

Simplify the sum.

$\small 1+3+5+...+(2n-1)$

Possible Answers:

$\small \frac{n(n+1)}{2}$

$\small n^2$

$\small \small \frac{(n-1)(n)}{2}$

$\small (n-1)^2$

$\small (n+1)^2$

Correct answer:

$\small n^2$

Explanation:

The answer is $\small n^2$ . Try this for $\small n=1,2,3,4$ :

$\small 1=1^2$

$\small 1+3=4=2^2$

$\small 1+3+5=9=3^2$

$\small 1+3+5+7=16=4^2$

This can be proven more generally using a proof technique called mathematical induction, which you will most likely not learn in high school.

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Example Question #31 : Sequences And Series

In case you are not familiar with summation notation, note that: $\sum_{i=1}^{3}\frac{1}{2}=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$

Given the series above, what is the value of $\sum_{i=1}^{6}\frac{1}{2}$ ?

Possible Answers:

4.5

Correct answer:

Explanation:

Since the upper bound of the iterator is and the initial value is , we need add one-half, the summand, six times.

This results in the following arithmetic.

$\\ \sum_{i=1}^{6}\frac{1}{2}\\ \\=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\\ \\= \frac{6}{2}\\ \\=3$

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Example Question #1 : Partial Sums Of Series

In case you are not familiar with summation notation, note that: $\sum_{i=1}^{3}(\frac{1}{2})^{i}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}$

What is the value of $\sum_{i=0}^{3}\left(\frac{1}{2}\right)^{i}$ ?

Possible Answers:

$\frac{5}{8}$

$\frac{15}{8}$

$\frac{31}{16}$

0.825

Correct answer:

$\frac{15}{8}$

Explanation:

Because the iterator starts at , we first have a $\frac{1}{2}^0 = 1$ .

Now expanding the summation to show the step by step process involved in answering the question we get,

$\\ \sum_{i=0}^{3}\left(\frac{1}{2}\right)^{i}=\left(\frac{1}{2}\right)^0+\left(\frac{1}{2}\right)^1+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3\\ \\=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\\ \\=\frac{8}{8}+\frac{4}{8}+\frac{2}{8}+\frac{1}{8}\\ \\=\frac{15}{8}$

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Example Question #1 : Sums Of Infinite Series

Find the value for $\sum_{i=0}^{\infty} (\frac{1}{2})^i$

Possible Answers:

1.25

1.5

Correct answer:

Explanation:

To best understand, let's write out the series. So

$\sum_{i=0}^{\infty} (\frac{1}{2})^i = 1+1/2+(1/2)^2 + (1/2)^3+...+(1/2)^\infty$

We can see this is an infinite geometric series with each successive term being multiplied by $\frac{1}{2}$ .

A definition you may wish to remember is

$1+r+r^2+r^3+... +r^\infty= 1/(1-r)$ where stands for the common ratio between the numbers, which in this case is $\frac{1}{2}$ or . So we get

1/(1-r) = 1/(1-.5) = 1/.5 = 2

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Example Question #1 : Sums Of Infinite Series

Evaluate:

$1 + \frac{1}{7} + \frac{1}{49} + \frac{1}{343} +...$

Possible Answers:

$\frac{6}{5}$

$\frac{9}{8}$

$\frac{7}{6}$

The series does not converge.

$\frac{4}{3}$

Correct answer:

$\frac{7}{6}$

Explanation:

This is a geometric series whose first term is $a_{0} = 1$ and whose common ratio is $r = \frac{1}{7}$ . The sum of this series is:

$\frac{a_{0}}{1-r} = \frac{1}{1- \frac{1}{7} } = \frac{1}{\frac{7}{7}- \frac{1}{7}}= \frac{1}{\frac{6}{7}} =\frac{7}{6}$

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Example Question #3 : Sums Of Infinite Series

Evaluate:

$1 - \frac{1}{6} + \frac{1}{36} - \frac{1}{216} +...$

Possible Answers:

$\frac{6}{7}$

The series does not converge.

$\frac{2}{3}$

$\frac{4}{5}$

$\frac{7}{8}$

Correct answer:

$\frac{6}{7}$

Explanation:

This is a geometric series whose first term is $a_{0} = 1$ and whose common ratio is $r = - \frac{1}{6}$ . The sum of this series is:

$\frac{a_{0}}{1-r} = \frac{1}{1- \left (- \frac{1}{6} \right )} = \frac{1}{\frac{6}{6}+ \frac{1}{6}}= \frac{1}{\frac{7}{6}} =\frac{6}{7}$

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Example Question #1 : Sums Of Infinite Series

What is the sum of the following infinite series?

$S=2-\frac{1}{3}+1-\frac{1}{9} +\frac{1}{2}-\frac{1}{27}+\frac{1}{4}-\frac{1}{81}+\cdots$

Possible Answers:

$3\tfrac{5}{9}$

diverges

$3\tfrac{1}{2}$

$4\tfrac{1}{2}$

Correct answer:

$3\tfrac{1}{2}$

Explanation:

This series is not alternating - it is the mixture of two geometric series.

The first series has the positive terms.

$2+1+\frac{1}{2}+ \frac{1}{4} + \cdots = 2*\sum\limits_{i=0}^\infty (\frac{1}{2})^i = 2*(\frac{1}{1-\tfrac{1}{2}})=4$

The second series has the negative terms.

$-\frac{1}{3}- \frac{1}{9} -\frac{1}{27}- \cdots = -\frac{1}{3}*\sum\limits_{i=0}^\infty (\frac{1}{3})^i = -\frac{1}{3}*(\frac{1}{1-\tfrac{1}{3}})=-0.5$

The sum of these values is 3.5.

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Example Question #1 : Sums Of Infinite Series

What is the sum of the alternating series below?

$10 - 5 + \frac{5}{2} - \frac{5}{4}+\frac{5}{8}-\frac{5}{16}+\cdots$

Possible Answers:

$\frac{20}{3}$

$\frac{10}{3}$

7.5

Correct answer:

$\frac{20}{3}$

Explanation:

The alternating series follows a geometric pattern.

$10 - 5 + \frac{5}{2} - \frac{5}{4} + \cdots = 10*\sum\limits_{i=0}^\infty(-\frac{1}{2})^i$

We can evaluate the geometric series from the formula.

$10*\sum\limits_{i=0}^\infty(-\frac{1}{2})^i = 10*\frac{1}{1+\tfrac{1}{2}} = 10*\frac{2}{3} =\frac{20}{3}$

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Precalculus : Sequences and Series

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #31 : Sequences And Series

Example Question #1 : Partial Sums Of Series

Example Question #1 : Partial Sums Of Series

Example Question #31 : Sequences And Series

Example Question #1 : Partial Sums Of Series

Example Question #1 : Sums Of Infinite Series

Example Question #1 : Sums Of Infinite Series

Example Question #3 : Sums Of Infinite Series

Example Question #1 : Sums Of Infinite Series

Example Question #1 : Sums Of Infinite Series

All Precalculus Resources

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