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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

By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
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Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
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Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
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Jhovie
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Charmaine Reply
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Daniel
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emmanuel
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Miranda
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Propessor Reply
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Miranda
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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