College Algebra 9th txtbk

SECTION 8–3 Arithmetic and Geometric Sequences 523
Now use Theorem 2 again, this time with n 17.
a 17 a 1 r 16 1(4 19 ) 16 4 169 11.758
MATCHED PROBLEM 2 (A) If the first and fifteenth terms of an arithmetic sequence are 5 and 23, respectively,
find the seventy-third term of the sequence.
1
(B) Find the eighth term of the geometric sequence
64 , 1
32 , 1 16 , . . . .
Z Developing Sum Formulas for Finite Arithmetic Series
If a 1 , a 2 , a 3 , . . . , a n is a finite arithmetic sequence, then the corresponding series
a 1 a 2 a 3 . . . a n is called an arithmetic series. We will derive two simple and
very useful formulas for the sum of an arithmetic series. Let d be the common difference
of the arithmetic sequence a 1 , a 2 , a 3 , . . . , a n and let S n denote the sum of the
series a 1 a 2 a 3 . . . a n .
Then
S n a 1 (a 1 d) . . . [a 1 (n 2)d] [a 1 (n 1)d]
Reversing the order of the sum, we obtain
S n [a 1 (n 1)d] [a 1 (n 2)d] . . . (a 1 d ) a 1
Adding the left sides of these two equations and corresponding elements of the right sides,
we see that
2S n [2a 1 (n 1)d] [2a 1 (n 1)d] . . . [2a 1 (n 1)d]
n[2a 1 (n 1)d]
This can be restated as in Theorem 3.
Z THEOREM 3 Sum of an Arithmetic Series—First Form
S n n 2 [2a 1 (n 1)d]
By replacing a 1 (n 1)d with a n , we obtain a second useful formula for the sum.
Z THEOREM 4 Sum of an Arithmetic Series—Second Form
S n n 2 (a 1 a n )
The proof of the first sum formula by mathematical induction is left as an exercise (see
Problem 68 in Exercises 8-3).