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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

the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
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
more than 6000
Robert
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Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
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
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
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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