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Algebra II : Sigma Notation

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Example Questions

Example Question #1 : Sigma Notation

$\sum_{n=1}^{5}2n-3$

What is the sum of the series?

Possible Answers:

Correct answer:

Explanation:

$\sum_{n=1}^{5}2n-3$

The number on the bottom is the first value we plug in for n. We keep substituting higher and higher integer values of n until we get to the top number (in this case, 5).

Here's what it looks like:

[2(1)-3]+[2(2)-3]+[2(3)-3]+[2(4)-3]+[2(5)-3]

[2-3]+[4-3]+[6-3]+[8-3]+[10-3]

-1+1+3+5+7

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Example Question #21 : Mathematical Relationships And Basic Graphs

$\sum_{n=1}^{\infty }\frac{1}{2^n}$

What is the sum of the series?

Possible Answers:

$-\frac{1}{2}$

$\frac{1}{2}$

Correct answer:

Explanation:

Recall the formula for the sum of an infinite geometric series:

$S = \frac{a}{1-r}$

Of course, this formula only works if the series is geometric with a common ratio between -1 and 1.

While the sigma notation calls for substituting an infinite number of values for n, let's just substitute a few to see if we can find a pattern:

$\frac{1}{2^1}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+...$

$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...$

From this, we can see that we have a geometric series with a common ratio of 1/2 and a first term of 1/2.

$S = \frac{\frac{1}{2}}{1-\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{1}{2}}=1$

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Example Question #1 : Sigma Notation

Solve for if $x = \sum i^{2}$ for i = 1 to i = 5

Possible Answers:

Correct answer:

Explanation:

For summations, we evaluate the expression at each value of , then add all of the results together.

For this problem, we are working from i = 1 to i = 5 .

i = 1 : i^2 = 1

i = 2 : 2^2=4

i = 3 : 3^2=9

i = 4 : 4^2=16

i = 5 : 5^2=25

Then adding everything up, we get 1 + 4 + 9 + 16 + 25 = 55

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Example Question #1 : Sigma Notation

Evaluate the following summation:

$\sum_{n=1}^{5}\frac{n^2+4}{2n}$

Possible Answers:

$\frac{4}{3}$

$\frac{29}{10}$

$\frac{211}{42}$

$\frac{181}{15}$

$\frac{79}{16}$

Correct answer:

$\frac{181}{15}$

Explanation:

According to the notation in the problem, we are told to sum the results obtained by evaluating the equation at each integer between the numbers below and above the sigma. For the sigma notation of this problem in particular, this means we start by plugging 1 into our equation, and then add the results obtained from plugging in 2, and then 3, and then 4, stopping after we add the result obatined from plugging 5 into the equation, as this is the number on top of sigma at which we stop the summation. This process is worked out below:

$\sum_{n=1}^{5}\frac{n^2+4}{2n}=\frac{1^2+4}{2(1)}+\frac{2^2+4}{2(2)}+\frac{3^2+4}{2(3)}+\frac{4^2+4}{2(4)}+\frac{5^2+4}{2(5)}$

$=\frac{5}{2}+2+\frac{13}{6}+\frac{5}{2}+\frac{29}{10}=\frac{75}{30}+\frac{60}{30}+\frac{65}{30}+\frac{75}{30}+\frac{87}{30}$

$=\frac{362}{30}=\frac{181}{15}$

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Example Question #1 : Sigma Notation

Evaluate: $\sum_{n=1}^{5} n$

Possible Answers:

120

Correct answer:

Explanation:

The summation starts from 1 and ends at 5. This can be rewritten as:

$\sum_{n=1}^{5} n =1+2+3+4+5=15$

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Example Question #1 : Sigma Notation

Evaluate: $\sum_{n=1}^{5} n^2$

Possible Answers:

14400

225

Correct answer:

Explanation:

The summation starts from 1 and ends at 5. Rewrite the summation sign:

$\sum_{n=1}^{5} n^2=1^2+2^2+3^2+4^2+5^2= 1+4+9+16+25=55$

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Example Question #1 : Sigma Notation

Evaluate: $\sum_{n=1}^{5} (-1)^n (2^n)$

Possible Answers:

Correct answer:

Explanation:

Rewrite the summation starting from 1 to 5 and add the terms.

$\sum_{n=1}^{5} (-1)^n (2^n)=$

(-1)^1 (2^1)+(-1)^2 (2^2)+(-1)^3 (2^3)+(-1)^4 (2^4)+(-1)^5 (2^5)

=-2+4-8+16-32

=-22

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Example Question #1 : Sigma Notation

Evaluate: $\sum_{n=0}^{3} \ln(n)$

Possible Answers:

$\ln(9)$

$\ln(2)+\ln(3)$

$\text{Does not exist}$

e^6

$\ln(7)$

Correct answer:

$\text{Does not exist}$

Explanation:

The natural log domain is only valid for values greater than zero. Therefore, the solution does not exist.

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Example Question #1 : Sigma Notation

The Fibonacci sequence is given as

a_1=1

a_2=1

a_3=2

a_4=3

a_5=5

a_6=8

a_7=13

Find

$\sum_{i=1}^{4}a_i$ .

Possible Answers:

Correct answer:

Explanation:

$\sum_{i=1}^{4}a_i$ is defined to be the sum of the a_i terms from $1\leq i\leq 4$ for $i\epsilon \mathbb{Z}$ .

In other words,

$\\ \sum_{i=1}^{4}a_i \\ \\= a_1+a_2+a_3+a_4\\ \\=1+1+2+3\\ \\=7$

As such, the asnwer is .

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Example Question #32 : Mathematical Relationships And Basic Graphs

Calculate: $\sum_{1}^{4} n+2$

Possible Answers:

Correct answer:

Explanation:

The sigma notation indicates summation starting from 1 and ends at 4.

Substitute in order of the iteration.

$\sum_{1}^{4} n+2$

= (1+2)+(2+2)+(3+2)+(4+2)

= 3+4+5+6

$\sum_{1}^{4} n+2 = 18$

The correct answer is .

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Algebra II : Sigma Notation

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Sigma Notation

Example Question #21 : Mathematical Relationships And Basic Graphs

Example Question #1 : Sigma Notation

Example Question #1 : Sigma Notation

Example Question #1 : Sigma Notation

Example Question #1 : Sigma Notation

Example Question #1 : Sigma Notation

Example Question #1 : Sigma Notation

Example Question #1 : Sigma Notation

Example Question #32 : Mathematical Relationships And Basic Graphs

All Algebra II Resources

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