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What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?

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Describe how linear functions and arithmetic sequences are similar. How are they different?

Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.

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Algebraic

For the following exercises, find the common difference for the arithmetic sequence provided.

{ 5 , 11 , 17 , 23 , 29 , ... }

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{ 0 , 1 2 , 1 , 3 2 , 2 , ... }

The common difference is 1 2

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For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.

{ 11.4 , 9.3 , 7.2 , 5.1 , 3 , ... }

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{ 4 , 16 , 64 , 256 , 1024 , ... }

The sequence is not arithmetic because 16 4 64 16.

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For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.

a 1 = 0 , d = 2 3

0 , 2 3 , 4 3 , 2 , 8 3

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For the following exercises, write the first five terms of the arithmetic series given two terms.

a 1 = 17 , a 7 = 31

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a 13 = 60 , a 33 = 160

0 , 5 , 10 , 15 , 20

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For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.

First term is 3, common difference is 4, find the 5 th term.

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First term is 4, common difference is 5, find the 4 th term.

a 4 = 19

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First term is 5, common difference is 6, find the 8 th term.

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First term is 6, common difference is 7, find the 6 th term.

a 6 = 41

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First term is 7, common difference is 8, find the 7 th term.

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For the following exercises, find the first term given two terms from an arithmetic sequence.

Find the first term or a 1 of an arithmetic sequence if a 6 = 12 and a 14 = 28.

a 1 = 2

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Find the first term or a 1 of an arithmetic sequence if a 7 = 21 and a 15 = 42.

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Find the first term or a 1 of an arithmetic sequence if a 8 = 40 and a 23 = 115.

a 1 = 5

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Find the first term or a 1 of an arithmetic sequence if a 9 = 54 and a 17 = 102.

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Find the first term or a 1 of an arithmetic sequence if a 11 = 11 and a 21 = 16.

a 1 = 6

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For the following exercises, find the specified term given two terms from an arithmetic sequence.

a 1 = 33 and a 7 = 15. Find a 4 .

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a 3 = 17.1 and a 10 = 15.7. Find a 21 .

a 21 = 13.5

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For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.

a 1 = 39 ;   a n = a n 1 3

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a 1 = 19 ;   a n = a n 1 1.4

19 , 20.4 , 21.8 , 23.2 , 24.6

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For the following exercises, write a recursive formula for each arithmetic sequence.

a n = { 40 , 60 , 80 , ... }

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a n = { 17 , 26 , 35 , ... }

a 1 = 17 ;   a n = a n 1 + 9 n 2

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a n = { 1 , 2 , 5 , ... }

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a n = { 12 , 17 , 22 , ... }

a 1 = 12 ;   a n = a n 1 + 5 n 2

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a n = { 15 , 7 , 1 , ... }

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a n = { 8.9 , 10.3 , 11.7 , ... }

a 1 = 8.9 ;   a n = a n 1 + 1.4 n 2

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a n = { 0.52 , 1.02 , 1.52 , ... }

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a n = { 1 5 , 9 20 , 7 10 , ... }

a 1 = 1 5 ;   a n = a n 1 + 1 4 n 2

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a n = { 1 2 , 5 4 , 2 , ... }

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a n = { 1 6 , 11 12 , 2 , ... }

1 = 1 6 ;   a n = a n 1 13 12 n 2

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For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.

a n = { 7 4 1 ... } ; Find the 17 th term.

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a n = { 4 11 18 ... } ; Find the 14 th term.

a 1 = 4 ;   a n = a n 1 + 7 ;   a 14 = 95

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a n = { 2 6 10 ... } ; Find the 12 th term.

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Questions & Answers

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
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
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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