# 9.2 Arithmetic sequences  (Page 3/8)

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Write a recursive formula for the arithmetic sequence.

## Using explicit formulas for arithmetic sequences

We can think of an arithmetic sequence    as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.

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

To find the y -intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.

The common difference is $-50$ , so the sequence represents a linear function with a slope of $-50$ . To find the $y$ -intercept, we subtract $-50$ from $200:\text{\hspace{0.17em}}200-\left(-50\right)=200+50=250$ . You can also find the $y$ -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in [link] .

Recall the slope-intercept form of a line is $\text{\hspace{0.17em}}y=mx+b.\text{\hspace{0.17em}}$ When dealing with sequences, we use ${a}_{n}$ in place of $y$ and $n$ in place of $x.\text{\hspace{0.17em}}$ If we know the slope and vertical intercept of the function, we can substitute them for $m$ and $b$ in the slope-intercept form of a line. Substituting $\text{\hspace{0.17em}}-50\text{\hspace{0.17em}}$ for the slope and $250$ for the vertical intercept, we get the following equation:

${a}_{n}=-50n+250$

We do not need to find the vertical intercept to write an explicit formula    for an arithmetic sequence. Another explicit formula for this sequence is ${a}_{n}=200-50\left(n-1\right)$ , which simplifies to $\text{\hspace{0.17em}}{a}_{n}=-50n+250.$

## Explicit formula for an arithmetic sequence

An explicit formula for the $n\text{th}$ term of an arithmetic sequence is given by

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

Given the first several terms for an arithmetic sequence, write an explicit formula.

1. Find the common difference, ${a}_{2}-{a}_{1}.$
2. Substitute the common difference and the first term into ${a}_{n}={a}_{1}+d\left(n-1\right).$

## Writing the n Th term explicit formula for an arithmetic sequence

Write an explicit formula for the arithmetic sequence.

The common difference can be found by subtracting the first term from the second term.

$\begin{array}{ll}d\hfill & ={a}_{2}-{a}_{1}\hfill \\ \hfill & =12-2\hfill \\ \hfill & =10\hfill \end{array}$

The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.

$\begin{array}{l}{a}_{n}=2+10\left(n-1\right)\hfill \\ {a}_{n}=10n-8\hfill \end{array}$

Write an explicit formula for the following arithmetic sequence.

$\left\{50,47,44,41,\dots \right\}$

${a}_{n}=53-3n$

## Finding the number of terms in a finite arithmetic sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

1. Find the common difference $d.$
2. Substitute the common difference and the first term into ${a}_{n}={a}_{1}+d\left(n–1\right).$
3. Substitute the last term for ${a}_{n}$ and solve for $n.$

## Finding the number of terms in a finite arithmetic sequence

Find the number of terms in the finite arithmetic sequence .

The common difference can be found by subtracting the first term from the second term.

$1-8=-7$

The common difference is $-7$ . Substitute the common difference and the initial term of the sequence into the $n\text{th}$ term formula and simplify.

$\begin{array}{l}{a}_{n}={a}_{1}+d\left(n-1\right)\hfill \\ {a}_{n}=8+-7\left(n-1\right)\hfill \\ {a}_{n}=15-7n\hfill \end{array}$

Substitute $-41$ for ${a}_{n}$ and solve for $n$

$\begin{array}{l}-41=15-7n\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}8=n\hfill \end{array}$

There are eight terms in the sequence.

#### 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
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salma
Commplementary angles
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Sherica
im all ears I need to learn
Sherica
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Tamia
hii
Uday
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salma
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Hello
opoku
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Ali
greetings from Iran
Ali
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Bach
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Nharnhar