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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

show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
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
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Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
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
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Harshika Reply
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Harshika
find the subring of gaussian integers?
Rofiqul
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Mark
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find the value of 2x=32
Felix Reply
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corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
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Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
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Opoku
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Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
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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
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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