High School Math : Understanding Arithmetic and Geometric Series

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Sequences And Series

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

Possible Answers:

\displaystyle \frac{3}{5}

\displaystyle 3

\displaystyle \frac{2}{5}

None of the other answers are correct.

\displaystyle \frac{5}{3}

Correct answer:

\displaystyle \frac{3}{5}

Explanation:

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

\displaystyle 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}

Example Question #1 : Arithmetic And Geometric 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:

220

110

55

105

210

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

\displaystyle 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:

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

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

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

\displaystyle a_1+3d=-20

\displaystyle a_1+7d=-10

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

\displaystyle -4d=-10.

This means that d = 2.5.

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

\displaystyle a_1+7(2.5)=-10

\displaystyle a_1=-27.5

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

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

The answer is 220.

Example Question #1 : Sequences And Series

Find the sum, if possible:

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

Possible Answers:

\displaystyle 9

\displaystyle 1

\displaystyle \frac{9}{2}

\displaystyle \frac{3}{2}

\displaystyle 3

Correct answer:

\displaystyle \frac{9}{2}

Explanation:

The formula for the summation of an infinite geometric series is

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

where \displaystyle a_1 is the first term in the series and \displaystyle r 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:

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

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

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

Example Question #2 : Sequences And Series

Find the sum, if possible:

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

Possible Answers:

\displaystyle \frac{34}{7}

\displaystyle \frac{30}{7}

\displaystyle \frac{26}{7}

\displaystyle \frac{32}{7}

\displaystyle \frac{28}{7}

Correct answer:

\displaystyle \frac{32}{7}

Explanation:

The formula for the summation of an infinite geometric series is

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

where \displaystyle a_1 is the first term in the series and \displaystyle r 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:

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

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

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

Example Question #2 : Sequences And Series

Find the sum, if possible:

\displaystyle 1-3+9-27+...

Possible Answers:

\displaystyle 6725

\displaystyle 7625

\displaystyle 5267

No solution

\displaystyle 2765

Correct answer:

No solution

Explanation:

The formula for the summation of an infinite geometric series is

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

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

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

Since \displaystyle r=-3, there is no solution.

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