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a n = a 1 + ( n 1 ) d

Given the first term and the common difference of an arithmetic sequence, find the first several terms.

  1. Add the common difference to the first term to find the second term.
  2. Add the common difference to the second term to find the third term.
  3. Continue until all of the desired terms are identified.
  4. Write the terms separated by commas within brackets.

Writing terms of arithmetic sequences

Write the first five terms of the arithmetic sequence    with a 1 = 17 and d = 3 .

Adding 3 is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term.

The first five terms are { 17 , 14 , 11 , 8 , 5 }

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List the first five terms of the arithmetic sequence with a 1 = 1 and d = 5 .

{ 1 ,   6 ,   11 ,   16 ,   21 }

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Given any the first term and any other term in an arithmetic sequence, find a given term.

  1. Substitute the values given for a 1 , a n , n into the formula a n = a 1 + ( n 1 ) d to solve for d .
  2. Find a given term by substituting the appropriate values for a 1 , n , and d into the formula a n = a 1 + ( n 1 ) d .

Writing terms of arithmetic sequences

Given a 1 = 8 and a 4 = 14 , find a 5 .

The sequence can be written in terms of the initial term 8 and the common difference d .

{ 8 , 8 + d , 8 + 2 d , 8 + 3 d }

We know the fourth term equals 14; we know the fourth term has the form a 1 + 3 d = 8 + 3 d .

We can find the common difference d .

a n = a 1 + ( n 1 ) d a 4 = a 1 + 3 d a 4 = 8 + 3 d Write the fourth term of the sequence in terms of   a 1   and   d . 14 = 8 + 3 d Substitute   14   for   a 4 .   d = 2 Solve for the common difference .

Find the fifth term by adding the common difference to the fourth term.

a 5 = a 4 + 2 = 16
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Given a 3 = 7 and a 5 = 17 , find a 2 .

a 2 = 2

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Using recursive formulas for arithmetic sequences

Some arithmetic sequences are defined in terms of the previous term using a recursive formula    . The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.

a n = a n 1 + d n 2

Recursive formula for an arithmetic sequence

The recursive formula for an arithmetic sequence with common difference d is:

a n = a n 1 + d n 2

Given an arithmetic sequence, write its recursive formula.

  1. Subtract any term from the subsequent term to find the common difference.
  2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.

Writing a recursive formula for an arithmetic sequence

Write a recursive formula    for the arithmetic sequence    .

{ 18 7 4 15 26 , … }

The first term is given as −18 . The common difference can be found by subtracting the first term from the second term.

d = −7 ( −18 ) = 11

Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.

a 1 = 18 a n = a n 1 + 11 ,  for  n 2
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Do we have to subtract the first term from the second term to find the common difference?

No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.

Questions & Answers

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
a²=4
Roy 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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