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Find the number of terms in the finite arithmetic sequence.

{ 6 11 16 ... 56 }

There are 11 terms in the sequence.

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Solving application problems with arithmetic sequences

In many application problems, it often makes sense to use an initial term of a 0 instead of a 1 . In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:

a n = a 0 + d n

Solving application problems with arithmetic sequences

A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.

  1. Write a formula for the child’s weekly allowance in a given year.
  2. What will the child’s allowance be when he is 16 years old?
  1. The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.

    Let A be the amount of the allowance and n be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:

    A n = 1 + 2 n
  2. We can find the number of years since age 5 by subtracting.

    16 5 = 11

    We are looking for the child’s allowance after 11 years. Substitute 11 into the formula to find the child’s allowance at age 16.

    A 11 = 1 + 2 ( 11 ) = 23

    The child’s allowance at age 16 will be $23 per week.

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A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?

The formula is T n = 10 + 4 n , and it will take her 42 minutes.

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Access this online resource for additional instruction and practice with arithmetic sequences.

Key equations

recursive formula for nth term of an arithmetic sequence a n = a n 1 + d n 2
explicit formula for nth term of an arithmetic sequence a n = a 1 + d ( n 1 )

Key concepts

  • An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
  • The constant between two consecutive terms is called the common difference.
  • The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See [link] .
  • The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See [link] and [link] .
  • A recursive formula for an arithmetic sequence with common difference d is given by a n = a n 1 + d , n 2. See [link] .
  • As with any recursive formula, the initial term of the sequence must be given.
  • An explicit formula for an arithmetic sequence with common difference d is given by a n = a 1 + d ( n 1 ) . See [link] .
  • An explicit formula can be used to find the number of terms in a sequence. See [link] .
  • In application problems, we sometimes alter the explicit formula slightly to a n = a 0 + d n . See [link] .

Section exercises

Verbal

What is an arithmetic sequence?

A sequence where each successive term of the sequence increases (or decreases) by a constant value.

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How is the common difference of an arithmetic sequence found?

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How do we determine whether a sequence is arithmetic?

We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.

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Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-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
Alfred Reply
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?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
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
Adu
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
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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
kinnecy Reply
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.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
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
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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
Idrissa Reply
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
Practice Key Terms 2

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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