# 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.

#### Questions & Answers

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
sure. what is your question?
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?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar