<< Chapter < Page Chapter >> Page >

Writing the terms of an alternating sequence defined by an explicit formula

Write the first five terms of the sequence.

a n = ( 1 ) n n 2 n + 1

Substitute n = 1 , n = 2 , and so on in the formula.

n = 1 a 1 = ( 1 ) 1 2 2 1 + 1 = 1 2 n = 2 a 2 = ( 1 ) 2 2 2 2 + 1 = 4 3 n = 3 a 3 = ( 1 ) 3 3 2 3 + 1 = 9 4 n = 4 a 4 = ( 1 ) 4 4 2 4 + 1 = 16 5 n = 5 a 5 = ( 1 ) 5 5 2 5 + 1 = 25 6

The first five terms are { 1 2 , 4 3 ,− 9 4 , 16 5 ,− 25 6 } .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

In [link] , does the (–1) to the power of n account for the oscillations of signs?

Yes, the power might be n , n + 1 , n 1 , and so on, but any odd powers will result in a negative term, and any even power will result in a positive term.

Write the first five terms of the sequence:

a n = 4 n ( 2 ) n

The first five terms are { 2 ,   2 ,   3 2 ,   1 ,   5 8 } .

Got questions? Get instant answers now!

Investigating piecewise explicit formulas

We’ve learned that sequences are functions whose domain is over the positive integers. This is true for other types of functions, including some piecewise functions . Recall that a piecewise function is a function defined by multiple subsections. A different formula might represent each individual subsection.

Given an explicit formula for a piecewise function, write the first n terms of a sequence

  1. Identify the formula to which n = 1 applies.
  2. To find the first term, a 1 , use n = 1 in the appropriate formula.
  3. Identify the formula to which n = 2 applies.
  4. To find the second term, a 2 , use n = 2 in the appropriate formula.
  5. Continue in the same manner until you have identified all n terms.

Writing the terms of a sequence defined by a piecewise explicit formula

Write the first six terms of the sequence.

a n = { n 2 if  n  is not divisible by 3 n 3 if  n  is divisible by 3

Substitute n = 1 , n = 2 , and so on in the appropriate formula. Use n 2 when n is not a multiple of 3. Use n 3 when n is a multiple of 3.

a 1 = 1 2 = 1 1 is not a multiple of 3 .  Use  n 2 . a 2 = 2 2 = 4 2 is not a multiple of 3 .  Use  n 2 . a 3 = 3 3 = 1 3 is a multiple of 3 .  Use  n 3 . a 4 = 4 2 = 16 4 is not a multiple of 3 .  Use  n 2 . a 5 = 5 2 = 25 5 is not a multiple of 3 .  Use  n 2 . a 6 = 6 3 = 2 6 is a multiple of 3 .  Use  n 3 .

The first six terms are { 1 ,   4 ,   1 ,   16 ,   25 ,   2 } .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write the first six terms of the sequence.

a n = { 2 n 3 if  n  is odd 5 n 2 if  n  is even

The first six terms are { 2 ,   5 ,   54 ,   10 ,   250 ,   15 } .

Got questions? Get instant answers now!

Finding an explicit formula

Thus far, we have been given the explicit formula and asked to find a number of terms of the sequence. Sometimes, the explicit formula for the n th term of a sequence is not given. Instead, we are given several terms from the sequence. When this happens, we can work in reverse to find an explicit formula from the first few terms of a sequence. The key to finding an explicit formula is to look for a pattern in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases.

Given the first few terms of a sequence, find an explicit formula for the sequence.

  1. Look for a pattern among the terms.
  2. If the terms are fractions, look for a separate pattern among the numerators and denominators.
  3. Look for a pattern among the signs of the terms.
  4. Write a formula for a n in terms of n . Test your formula for n = 1 ,   n = 2 , and n = 3.

Writing an explicit formula for the n Th term of a sequence

Write an explicit formula for the n th term of each sequence.

  1. { 2 11 , 3 13 , 4 15 , 5 17 , 6 19 , }
  2. { 2 25 , 2 125 , 2 625 , 2 3 , 125 , 2 15 , 625 , }
  3. { e 4 , e 5 , e 6 , e 7 , e 8 , }

Look for the pattern in each sequence.

  1. The terms alternate between positive and negative. We can use ( 1 ) n to make the terms alternate. The numerator can be represented by n + 1. The denominator can be represented by 2 n + 9.

    a n = ( 1 ) n ( n + 1 ) 2 n + 9

  2. The terms are all negative.

    So we know that the fraction is negative, the numerator is 2, and the denominator can be represented by 5 n + 1 .

    a n = 2 5 n + 1
  3. The terms are powers of e . For n = 1 , the first term is e 4 so the exponent must be n + 3.

    a n = e n + 3
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
Practice Key Terms 8

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask