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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

the sum of any two linear polynomial is what
Esther Reply
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Momo Reply
divide simplify each answer 25/3÷5/12
Momo
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austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
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Robert
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Zairen Reply
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Peighton Reply
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Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
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Bryssen Reply
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meena Reply
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Den Reply
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Barcenas Reply
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Jeffrey
Confunction Identity
Barcenas
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meena
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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