# 9.2 Arithmetic sequences  (Page 2/8)

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${a}_{n}={a}_{1}+\left(n-1\right)d$

Given the first term and the common difference of an arithmetic sequence, find the first several terms.

1. Add the common difference to the first term to find the second term.
2. Add the common difference to the second term to find the third term.
3. Continue until all of the desired terms are identified.
4. Write the terms separated by commas within brackets.

## Writing terms of arithmetic sequences

Write the first five terms of the arithmetic sequence    with ${a}_{1}=17$ and $d=-3$ .

Adding $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.

The first five terms are $\text{\hspace{0.17em}}\left\{17,\text{\hspace{0.17em}}14,\text{\hspace{0.17em}}11,\text{\hspace{0.17em}}8,\text{\hspace{0.17em}}5\right\}$

List the first five terms of the arithmetic sequence with ${a}_{1}=1$ and $d=5$ .

Given any the first term and any other term in an arithmetic sequence, find a given term.

1. Substitute the values given for ${a}_{1},{a}_{n},n$ into the formula $\text{\hspace{0.17em}}{a}_{n}={a}_{1}+\left(n-1\right)d\text{\hspace{0.17em}}$ to solve for $\text{\hspace{0.17em}}d.$
2. Find a given term by substituting the appropriate values for $\text{\hspace{0.17em}}{a}_{1},n,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ into the formula ${a}_{n}={a}_{1}+\left(n-1\right)d.$

## Writing terms of arithmetic sequences

Given ${a}_{1}=8$ and ${a}_{4}=14$ , find ${a}_{5}$ .

The sequence can be written in terms of the initial term 8 and the common difference $d$ .

$\left\{8,8+d,8+2d,8+3d\right\}$

We know the fourth term equals 14; we know the fourth term has the form ${a}_{1}+3d=8+3d$ .

We can find the common difference $d$ .

Find the fifth term by adding the common difference to the fourth term.

${a}_{5}={a}_{4}+2=16$

Given ${a}_{3}=7$ and ${a}_{5}=17$ , find ${a}_{2}$ .

${a}_{2}=2$

## Using recursive formulas for arithmetic sequences

Some arithmetic sequences are defined in terms of the previous term using a recursive formula    . The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

$\begin{array}{lllll}{a}_{n}={a}_{n-1}+d\hfill & \hfill & \hfill & \hfill & n\ge 2\hfill \end{array}$

## Recursive formula for an arithmetic sequence

The recursive formula for an arithmetic sequence with common difference $d$ is:

$\begin{array}{lllll}{a}_{n}={a}_{n-1}+d\hfill & \hfill & \hfill & \hfill & n\ge 2\hfill \end{array}$

Given an arithmetic sequence, write its recursive formula.

1. Subtract any term from the subsequent term to find the common difference.
2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.

## Writing a recursive formula for an arithmetic sequence

Write a recursive formula    for the arithmetic sequence    .

The first term is given as $-18$ . The common difference can be found by subtracting the first term from the second term.

$d=-7-\left(-18\right)=11$

Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.

Do we have to subtract the first term from the second term to find the common difference?

No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali