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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

how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
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
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
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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