The general term is Now find
Let
Multiply.
Subtract. NOW TRYOBJECTIVE 3 Use an arithmetic sequence in an application.
Applying an Arithmetic SequenceLeonid Bekker’s uncle decides to start a fund for Leonid’s education. He makes an
initial contribution of $3000 and deposits an additional $500 each month. Thus, after
one month the fund will have How much will it have after
24 months? (Disregard any interest.)
After nmonths, the fund will contain
Use an arithmetic sequence.To find the amount in the fund after 24 months, find
Let
Multiply.
Add.The account will contain $15,000 (disregarding interest) after 24 months.
NOW TRYOBJECTIVE 4 Find any specified term or the number of terms of an arith-
metic sequence.The formula for the general term of an arithmetic sequence has
four variables: n, and d. If we know any three of these, the formula can be used
to find the value of the fourth variable.
Finding Specified Terms in SequencesEvaluate the indicated term for each arithmetic sequence.
(a) ;
Formula for
Let
Let
Multiply, and then add.(b) and ;
Any term can be found if and dare known. Use the formula for.
This gives a system of two equations in two variables, and d.
(1)a 1 + 10 d=- 10 (2)
a 1 + 4 d= 2
a 1
2 = a 1 + 4 d a 5 = 2 - 10 =a 1 + 10 d a 11 =- 10
a 5 = a 1 + 4 d a 11 =a 1 + 10 d
a 5 = a 1 + 15 - 12 d a 11 =a 1 + 111 - 12 d
a 1 an
a 5 = 2 a 11 =- 10 a 17
= 162
= - 6 + 141122 a 1 =-6, d=12.
a 15 =a 1 + 115 - 12 d n=15.
a (^) n=a 1 + 1 n- 12 d an
a 1 =-6,d= 12 a 15
EXAMPLE 5
an,a 1 ,
=15,000
= 3000 + 12,000
a 24 = 3000 + 5001242 n=24.
a 24.
an= 3000 + 500 n dollars.
$3000+ $500=$3500.
EXAMPLE 4
= 48
= 60 - 12
a 20 = 31202 - 12 n=20.
an= 3 n-12. a 20.
686 CHAPTER 12 Sequences and Series
NOW TRY
EXERCISE 3
Determine the general term of
the arithmetic sequence.
Then use the general term to
find a 20.
- 5, 0, 5, 10, 15,Á
NOW TRY ANSWERS
- an= 5 n- 10 ; a 20 = 90
NOW TRY
EXERCISE 4
Ginny Tiller is saving money
for her son’s college educa-
tion. She makes an initial
contribution of $1000 and de-
posits an additional $120 each
month for the next 96 months.
Disregarding interest, how
much money will be in the
account after 96 months?
- $12,520