# 13.2 Arithmetic sequences  (Page 2/8)

 Page 2 / 8
${a}_{n}={a}_{1}+\left(n-1\right)d$

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

exercise 1.2 solution b....isnt it lacking
Miiro Reply
I dnt get dis work well
john Reply
what is one-to-one function
Iwori Reply
what is the procedure in solving quadratic equetion at least 6?
Qhadz Reply
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
wisdom Reply
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
Gautam Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
favour Reply
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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