Multiply equation (2) by and add to equation (1) to eliminate
(1)
times (2)
Add.
Divide by.Now find by substituting for dinto either equation.
Let in (2).
Multiply.
Add 20.Use the formula for to find
Let.
Subtract.
Let=- 22 Simplify.
= 10 + 161 - 22 a 1 =10, d=-2.
=a 1 + 16 d
a 17 =a 1 + 117 - 12 d n= 17
an a 17.
a 1 = 10
a 1 - 20 =- 10
a 1 + 101 - 22 =- 10 d=- 2
a 1 - 2
d= - 2 - 6
- 6 d= 12
- a 1 - 10 d= 10 - 1
a 1 + 4 d= 2
- 1 a 1.
SECTION 12.2 Arithmetic Sequences 687
NOW TRY
EXERCISE 5
Evaluate the indicated term for
each arithmetic sequence.
(a) and ;
(b)a 7 = 25 and ;a 12 = 40 a 19
a 1 = 21 d=- 3 a 22NOW TRY ANSWERS
- (a)- 42 (b) 61
Multiply and
then add.NOW TRYFinding the Number of Terms in a SequenceEvaluate the number of terms in the arithmetic sequence.
Let nrepresent the number of terms in the sequence. Since
and use the formula for to find n.
Formula forLet
Distributive property
Simplify.
Divide by 6.The sequence has 11 terms. NOW TRY
OBJECTIVE 5 Find the sum of a specified number of terms of an arith-
metic sequence.To find a formula for the sum of the first nterms of a given
arithmetic sequence, we can write out the terms in two ways. We start with the first
term, and then with the last term. Then we add the terms in columns.
The right-hand side of this expression contains nterms, each equal to
Sn Divide by 2.
n
2
1 a 1 an 2
2 Sn= n 1 a 1 + an 2
a 1 +an.
2 Sn= 1 a 1 + an 2 + 1 a 1 +an 2 + 1 a 1 +an 2 + Á + 1 a 1 + an 2
Sn=an+ 1 an-d 2 + 1 an- 2 d 2 + Á+ 3 an- 1 n- 12 d 4
Sn=a 1 + 1 a 1 +d 2 + 1 a 1 + 2 d 2 + Á+ 3 a 1 + 1 n- 12 d 4
Sn
n= 11
66 = 6 n
52 =- 8 + 6 n- 6
52 = - 8 + 1 n- 12162 an=52,a 1 =-8, d=6.
a (^) n= a 1 + 1 n- 12 d an
d=- 2 - 1 - 82 = 6, an
an=52,a 1 =-8,
- 8, -2, 4, 10,Á, 52
NOW TRY EXAMPLE 6
EXERCISE 6
Evaluate the number of terms
in the arithmetic sequence.
1,4
3
,
5
3
, 2,Á, 11
- 31
Formula
for Sn