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Write a recursive formula for the arithmetic sequence.

{ 25   37   49   61   }

a 1 = 25 a n = a n 1 + 12 ,  for  n 2

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Using explicit formulas for arithmetic sequences

We can think of an arithmetic sequence    as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.

a n = a 1 + d ( n 1 )

To find the y -intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence.

A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50.

The common difference is 50 , so the sequence represents a linear function with a slope of 50 . To find the y -intercept, we subtract 50 from 200 : 200 ( 50 ) = 200 + 50 = 250 . You can also find the y -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in [link] .

Graph of the arithmetic sequence. The points form a negative line.

Recall the slope-intercept form of a line is y = m x + b . When dealing with sequences, we use a n in place of y and n in place of x . If we know the slope and vertical intercept of the function, we can substitute them for m and b in the slope-intercept form of a line. Substituting 50 for the slope and 250 for the vertical intercept, we get the following equation:

a n = 50 n + 250

We do not need to find the vertical intercept to write an explicit formula    for an arithmetic sequence. Another explicit formula for this sequence is a n = 200 50 ( n 1 ) , which simplifies to a n = 50 n + 250.

Explicit formula for an arithmetic sequence

An explicit formula for the n th term of an arithmetic sequence is given by

a n = a 1 + d ( n 1 )

Given the first several terms for an arithmetic sequence, write an explicit formula.

  1. Find the common difference, a 2 a 1 .
  2. Substitute the common difference and the first term into a n = a 1 + d ( n 1 ) .

Writing the n Th term explicit formula for an arithmetic sequence

Write an explicit formula for the arithmetic sequence.

{ 2 12 22 32 42 , … }

The common difference can be found by subtracting the first term from the second term.

d = a 2 a 1 = 12 2 = 10

The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.

a n = 2 + 10 ( n 1 ) a n = 10 n 8
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Write an explicit formula for the following arithmetic sequence.

{ 50 , 47 , 44 , 41 , }

a n = 53 3 n

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Finding the number of terms in a finite arithmetic sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

  1. Find the common difference d .
  2. Substitute the common difference and the first term into a n = a 1 + d ( n 1 ) .
  3. Substitute the last term for a n and solve for n .

Finding the number of terms in a finite arithmetic sequence

Find the number of terms in the finite arithmetic sequence .

{ 8 1 –6 ... –41 }

The common difference can be found by subtracting the first term from the second term.

1 8 = 7

The common difference is 7 . Substitute the common difference and the initial term of the sequence into the n th term formula and simplify.

a n = a 1 + d ( n 1 ) a n = 8 + 7 ( n 1 ) a n = 15 7 n

Substitute 41 for a n and solve for n

41 = 15 7 n 8 = n

There are eight terms in the sequence.

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Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
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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