Math - Understanding Arithmetic and Geometric Series

Find the sum, if possible:

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

$\frac{32}{7}$

$\frac{34}{7}$

$\frac{30}{7}$

$\frac{28}{7}$

$\frac{26}{7}$

Explanation

The formula for the summation of an infinite geometric series is

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

where 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:

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

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

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

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

$\frac{3}{5}$

$\frac{2}{5}$

$\frac{5}{3}$

None of the other answers are correct.

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 $r = -\frac{2}{3}$ :

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

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

220

110

210

105

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:

, 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:

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

Let us solve this system by subtracting the equation from the equation . The result of this subtraction is

This means that d = 2.5.

Using the equation , we can find the first term of the sequence.

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.

Find the sum, if possible:

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

$\frac{9}{2}$

$\frac{3}{2}$

Explanation

The formula for the summation of an infinite geometric series is

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

where 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.