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Write a recursive formula for the arithmetic sequence.

{ 25   37   49   61   }

a 1 = 25 a n = a n 1 + 12 ,  for  n 2

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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 ( n 1 )

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.

A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50.

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 : 200 ( 50 ) = 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] .

Graph of the arithmetic sequence. The points form a negative line.

Recall the slope-intercept form of a line is y = m x + b . When dealing with sequences, we use a n in place of y and n in place of x . 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 50 for the slope and 250 for the vertical intercept, we get the following equation:

a n = 50 n + 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 ( n 1 ) , which simplifies to a n = 50 n + 250.

Explicit formula for an arithmetic sequence

An explicit formula for the n th term of an arithmetic sequence is given by

a n = a 1 + d ( n 1 )

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 ( n 1 ) .

Writing the n Th term explicit formula for an arithmetic sequence

Write an explicit formula for the arithmetic sequence.

{ 2 12 22 32 42 , … }

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

d = a 2 a 1 = 12 2 = 10

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

a n = 2 + 10 ( n 1 ) a n = 10 n 8
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Write an explicit formula for the following arithmetic sequence.

{ 50 , 47 , 44 , 41 , }

a n = 53 3 n

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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 ( n 1 ) .
  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 .

{ 8 1 –6 ... –41 }

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 th term formula and simplify.

a n = a 1 + d ( n 1 ) a n = 8 + 7 ( n 1 ) a n = 15 7 n

Substitute 41 for a n and solve for n

41 = 15 7 n 8 = n

There are eight terms in the sequence.

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Questions & Answers

explain and give four Example hyperbolic function
Lukman Reply
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
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
yes i wantt to review
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
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
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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