Intermediate Algebra (11th edition)

Finding the Sum of the First nTerms of an Arithmetic Sequence

Evaluate the sum of the first five terms of the arithmetic sequence in which

Begin by evaluating and

Now evaluate the sum using and

Formula for

Substitute.

Add.

Multiply. NOW TRY

It is possible to express the sum of an arithmetic sequence in terms of and d,

the quantities that define the sequence. Since

and

by substituting the expression for into the expression for we obtain

Substitute for.

Combine like terms.

The summary box gives both of the alternative forms that may be used to find the

sum of the first nterms of an arithmetic sequence.

Sn

n

2

32 a 1 1 n 12 d 4.

Sn= an

n

2

1 a 1 + 3 a 1 + 1 n- 12 d 42

an Sn

Sn= an=a 1 + 1 n- 12 d,

n

2

1 a 1 + an 2

Sn a 1

= 5

=

5

2

122

S 5 =

5

2

1 - 3 + 52

Sn= Sn

n

2

1 a 1 +an 2

a 1 =-3,a 5 =5, n=5.

= - 3 = 5

a 1 = 2112 - 5 a 5 = 2152 - 5

a 1 a 5.

an= 2 n-5.

EXAMPLE 7

688 CHAPTER 12 Sequences and Series

NOW TRY
EXERCISE 7
Evaluate the sum of the first
seven terms of the arithmetic
sequence in which
an= 5 n-7.

NOW TRY ANSWER

91

Sum of the First nTerms of an Arithmetic Sequence

The sum of the first nterms of the arithmetic sequence with first term nth

term and common difference dis given by either formula.

or Sn

n

2

Sn 32 a 1 1 n 12 d 4

n

2

1 a 1 an 2

an,

a 1 ,