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Given a geometric sequence with a 2 = 4 and a 3 = 32 , find a 6 .

a 6 = 16 , 384

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

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

{ 2 10 50 250 ... }

The first term is 2. The common ratio can be found by dividing the second term by the first term.

10 2 = 5

The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.

a n = a 1 r ( n 1 ) a n = 2 5 n 1

The graph of this sequence in [link] shows an exponential pattern.

Graph of the geometric sequence.
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Write an explicit formula for the following geometric sequence.

{ –1 3 –9 27 ... }

a n = ( 3 ) n 1

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Solving application problems with geometric sequences

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of a 0 instead of a 1 . In these problems, we can alter the explicit formula slightly by using the following formula:

a n = a 0 r n

Solving application problems with geometric sequences

In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.

  1. Write a formula for the student population.
  2. Estimate the student population in 2020.
  1. The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.

    Let P be the student population and n be the number of years after 2013. Using the explicit formula for a geometric sequence we get

    P n   = 284 1.04 n
  2. We can find the number of years since 2013 by subtracting.

    2020 2013 = 7

    We are looking for the population after 7 years. We can substitute 7 for n to estimate the population in 2020.

    P 7 = 284 1.04 7 374

    The student population will be about 374 in 2020.

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A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.

  1. Write a formula for the number of hits.
  2. Estimate the number of hits in 5 weeks.

  1. P n   =   293 1.026 a n
  2. The number of hits will be about 333.

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Access these online resources for additional instruction and practice with geometric sequences.

Key equations

recursive formula for n t h term of a geometric sequence a n = r a n 1 , n 2
explicit formula for n t h term of a geometric sequence a n = a 1 r n 1

Key concepts

  • A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
  • The constant ratio between two consecutive terms is called the common ratio.
  • The common ratio can be found by dividing any term in the sequence by the previous term. See [link] .
  • The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. See [link] and [link] .
  • A recursive formula for a geometric sequence with common ratio r is given by a n = r a n 1 for n 2 .
  • As with any recursive formula, the initial term of the sequence must be given. See [link] .
  • An explicit formula for a geometric sequence with common ratio r is given by a n = a 1 r n 1 . See [link] .
  • In application problems, we sometimes alter the explicit formula slightly to a n = a 0 r n . See [link] .

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
bsc F. y algebra and trigonometry pepper 2
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
sorry
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
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
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Mark Reply
nothing up todat yet
Miranda
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jai
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Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
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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