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Precalculus : Partial Sums of Series

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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:

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

$\small (n-1)^2$

$\small (n+1)^2$

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

$\small n^2$

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 #3 : Partial Sums Of 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 #31 : Sequences And 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:

$\frac{31}{16}$

$\frac{5}{8}$

$\frac{15}{8}$

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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Precalculus : Partial Sums of Series

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Partial Sums Of Series

Example Question #1 : Partial Sums Of Series

Example Question #3 : Partial Sums Of Series

Example Question #31 : Sequences And Series

All Precalculus Resources

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