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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

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
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Shirley Reply
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Abdullahi
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Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
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Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
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Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
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Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
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
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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