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High School Math : Understanding Arithmetic and Geometric Series

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Example Questions

Example Question #1 : Sequences And Series

Evaluate: $1 - \frac{2}{3} + \frac{4}{9} - \frac{8}{27} + ...$

Possible Answers:

$\frac{2}{5}$

None of the other answers are correct.

$\frac{3}{5}$

$\frac{5}{3}$

Correct answer:

$\frac{3}{5}$

Explanation:

This sum can be determined using the formula for the sum of an infinite geometric series, with initial term $a_{0} = 1$ and common ratio $r = -\frac{2}{3}$ :

$S = \frac{a_{0}} {1-r} = \frac{1} {1- \left ( -\frac{2}{3}\right ) } = \frac{1} {1+ \frac{2}{3} }= \frac{1} { \frac{5}{3} } =\frac{3}{5}$

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Example Question #1 : Sequences And Series

The fourth term in an arithmetic sequence is -20, and the eighth term is -10. What is the hundredth term in the sequence?

Possible Answers:

210

220

110

105

Correct answer:

220

Explanation:

An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it.

Let $a_{n}$ denote the nth term of the sequence. Then the following formula can be used for arithmetic sequences in general:

a_n=a_1 +(n-1)d , where d is the common difference between two consecutive terms.

We are given the 4th and 8th terms in the sequence, so we can write the following equations:

a_4=a_1+(4-1)d=a_1+3d=-20

a_8=a_1+(8-1)d=a_1+7d=-10 .

We now have a system of two equations with two unknowns:

a_1+3d=-20

a_1+7d=-10

Let us solve this system by subtracting the equation a_1+7d=-10 from the equation a_1+3d=-20 . The result of this subtraction is

-4d=-10 .

This means that d = 2.5.

Using the equation a_1+7d=-10 , we can find the first term of the sequence.

a_1+7(2.5)=-10

a_1=-27.5

Ultimately, we are asked to find the hundredth term of the sequence.

$a_{100}=a_1+(100-1)d=-27.5+99(2.5)=220$

The answer is 220.

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Example Question #1 : Sequences And Series

Find the sum, if possible:

$3+1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...$

Possible Answers:

$\frac{9}{2}$

$\frac{3}{2}$

Correct answer:

$\frac{9}{2}$

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms. The key here is finding the rate, or pattern, between the terms. Because this is a geometric sequence, the rate is the constant by which each new term is multiplied.

Plugging in our values, we get:

$S = \frac{3}{1-\frac{1}{3}}$

$S=\frac{3}{\frac{2}{3}}$

$S = \frac{9}{2}$

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Example Question #2 : Sequences And Series

Find the sum, if possible:

$8-6+\frac{9}{2}-\frac{27}{8}+...$

Possible Answers:

$\frac{34}{7}$

$\frac{30}{7}$

$\frac{26}{7}$

$\frac{32}{7}$

$\frac{28}{7}$

Correct answer:

$\frac{32}{7}$

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms in a series

Because the terms switch sign, we know that the rate must be negative.

Plugging in our values, we get:

$S = \frac{8}{1-(\frac{-3}{4})}$

$S = \frac{8}{\frac{7}{4}}$

$S = \frac{32}{7}$

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Example Question #2 : Sequences And Series

Find the sum, if possible:

1-3+9-27+...

Possible Answers:

6725

7625

5267

No solution

2765

Correct answer:

No solution

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms in a series.

In order for an infinite geometric series to have a sum, needs to be greater than and less than , i.e. -1<r<1 .

Since r=-3 , there is no solution.

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High School Math : Understanding Arithmetic and Geometric Series

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Sequences And Series

Example Question #1 : Sequences And Series

Example Question #1 : Sequences And Series

Example Question #2 : Sequences And Series

Example Question #2 : Sequences And Series

All High School Math Resources

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