Finding Terms in a Series

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Questions 1 - 8

Indicate the first three terms of the following series:

$\sum_{c=3}^{10}(8-5c)$

Explanation

In the arithmetic series, the first terms can be found by plugging in , , and for .

Indicate the first three terms of the following series.

$\sum_{b=2}^{11} \frac{1}{2}(4)^{b-2}$

$\frac{1}{2}+2+8$

$\frac{1}{2}\cdot 2\cdot 8$

$\frac{1}{4}+4+16$

$\frac{1}{4}\cdot 4\cdot 16$

Not enough information

Explanation

The first terms can be found by substituting , , and in for .

$\frac{1}{2}(4)^{2-2}=\frac{1}{2}$

$\frac{1}{2}(4)^{3-2}=2$

$\frac{1}{2}(4)^{4-2}=8$

What are the first three terms in the series?

$\sum_{n=1}^{7}(4n-5)$

Explanation

To find the first three terms, replace with , , and .

The first three terms are , , and .

Find the first three terms in the series.

$\sum_{n=3}^{10}(8-5n)$

Explanation

To find the first three terms, replace with , , and .

The first three terms are , , and .

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

What is the fifteenth term in the sequence?

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, .

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

Indicate the first three terms of the following series:

$\sum_{x=1}^{7}(4x-5)$

Explanation

In the arithmetic series, the first terms can be found by plugging , , and into the equation.

What is the sixth term when $\left (\frac{1}{2}x-2 \right )^{10}$ is expanded?

Explanation

We will need to use the Binomial Theorem in order to solve this problem. Consider the expansion of , where n is an integer. The rth term of this expansion is given by the following formula:

$\binom{n}{r-1}a^{n-r+1}b^{r-1}$ ,

where $\binom{n}{r-1}$ is a combination. In general, if x and y are nonnegative integers such that x > y, then the combination of x and y is defined as follows: $\binom{x}{y}=_xC_y=\frac{x!}{(y)!(x-y)!}$ .

We are asked to find the sixth term of $\left (\frac{1}{2}x-2 \right )^{10}$ , which means that in this case r = 6 and n = 10. Also, we will let $a=\frac{1}{2}x$ and . We can now apply the Binomial Theorem to determine the sixth term, which is as follows: