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Precalculus : Pre-Calculus

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Example Questions

Example Question #11 : Sigma Notation

What is the proper sigma sum notation of the summation of $1+n+n^2+...+n^{99}+n^{100}$ ?

Possible Answers:

$\sum_{j=1}^{101}j^n$

$\sum_{j=100}^{0}n^j$

$\sum_{j=1}^{101}n^j$

$\su\sum_{j=0}^{100}n^j$

Correct answer:

$\su\sum_{j=0}^{100}n^j$

Explanation:

Given that the first term of the sequence is , we know that the first term of the summation must be n^0 , and thus the lower bound of summation must be equal to There is only one option with this qualification, and so we have our answer.

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Example Question #1 : Finding Terms In A Series

Consider the sequence: $2,\ 5,\ 8,\ 11,\ ...$

What is the fifteenth term in the sequence?

Possible Answers:

Correct answer:

Explanation:

The sequence can be described by the equation $\small 2+3(n-1)$ , where is the term in the sequence.

For the 15th term, n=15 .

$\small 2+3(n-1)=2+3(15-1)$

2+3(14)

2+42

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Example Question #1 : Terms In A Series

What is the sum of the first terms of an arithmetic series if the first term is , and the last term is ?

Possible Answers:

$\frac{271}{15}$

$\frac{181}{15}$

$\frac{315}{2}$

$\frac{317}{4}$

Correct answer:

$\frac{315}{2}$

Explanation:

Write the formula to find the arithmetic sum of a series where is the number of terms, a_1 is the first term, and a_n is the last term.

$S=\frac{n(a_1+a_n)}{2}$

Substitute the given values and solve for the sum.

$S=\frac{15(1+20)}{2}=\frac{15 \cdot21}{2}= \frac{315}{2}$

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Example Question #3 : Terms In A Series

Given the terms of the sequence $\small \small 0,1,1,2,3,5,...$ , what are the next two terms after $\small 5$ ?

Possible Answers:

$\small 7, 10$

5, 5

$\small 8, 12$

$\small 7, 13$

$\small 8, 13$

Correct answer:

$\small 8, 13$

Explanation:

The next two terms are $\small 8$ and $\small 13$ . This is the Fibonacci sequence where you start off with the terms $\small 0$ and $\small 1$ , and the next term is the sum of two previous terms. So then

$\small a_3=0+1=1$

$\small a_4=1+1=2$

$\small a_5=1+2=3$

$\small a_6 = 2+3=5$

$\small a_7=3+5=8$

$\small a_8 = 5+8=13$

$\small \small a_{9}=8+13=21$

and so on.

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Example Question #1 : Terms In A Series

What is the fifth term of the series 0.5+0.25+0.125+...

Possible Answers:

0.125125

$\frac{1}{16}$

$\frac{1}{32}$

0.1125

0.5125

Correct answer:

$\frac{1}{32}$

Explanation:

Let's try to see if this series is a geometric series.

We can divide adjacent terms to try and discover a multiplicative factor.

Doing this it seems the series proceeds with a common multiple of $\frac{1}{2}$ between each term.

Rewriting the series we get,

$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\left(\frac{1}{2}\right)^n$ .

When

$n=5\rightarrow \frac{1}{2^5}=\frac{1}{32}$ .

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Example Question #181 : Pre Calculus

What is the 9th term of the series that begins 2, 4, 8, 16...

Possible Answers:

256

1024

144

512

488

Correct answer:

512

Explanation:

In this geometric series, each number is created by multiplying the previous number by 2. You may also see that, because the first number is 2, it also becomes a list of powers of 2. The list is 2, 4, 8, 16, 32, 64, 128, 256, 512, where you can see that the 9th term is 512.

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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 #2 : Partial Sums Of Series

Simplify the sum.

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

Possible Answers:

$\small (n+1)^2$

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

$\small n^2$

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

$\small (n-1)^2$

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 #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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Precalculus : Pre-Calculus

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #11 : Sigma Notation

Example Question #1 : Finding Terms In A Series

Example Question #1 : Terms In A Series

Example Question #3 : Terms In A Series

Example Question #1 : Terms In A Series

Example Question #181 : Pre Calculus

Example Question #31 : Sequences And Series

Example Question #1 : Partial Sums Of Series

Example Question #2 : Partial Sums Of Series

Example Question #3 : Partial Sums Of Series

All Precalculus Resources

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