694 CHAPTER 12 Sequences and Series

Sum of the First nTerms of a Geometric Sequence

The sum of the first nterms of the geometric sequence with first term and

common ratio ris

Sn 1 r 12.

a 111 rn 2

1 r

a 1

If then

Multiplying the formula for by gives an alternative form.

Alternative form

Finding the Sum of the First nTerms of a Geometric

Sequence

Evaluate the sum of the first six terms of the geometric sequence with first term

and common ratio 3.

Formula for

Let

Simplify. NOW TRY

A series of the form

represents the sum of the first nterms of a geometric sequence having first term

a 1 = a#b^1 = aband common ratio b. The next example illustrates this form.

a

n

i 1

a#bi

=- 728

=

- 211 - 7292

- 2

S 6 = n=6,a 1 =-2, r=3.

- 211 - 362

1 - 3

Sn= Sn

a 111 - rn 2

1 - r

- 2

EXAMPLE 5

Sn=

a 111 - rn 2

1 - r

-^1

- 1

=

a 11 rn 12

r 1

• 1

Sn - 1

r= 1, Sn=a 1 + a 1 +a 1 + Á+ a 1 =na 1.

NOW TRY
EXERCISE 5
Evaluate the sum of the first
six terms of the geometric
sequence with first term 4
and common ratio 2.

NOW TRY ANSWERS

5. 252 6.7.75, or^314

NOW TRY

EXERCISE 6

Evaluate .a

5

i= 1

8 a

1

2

b

i

Evaluate Subtract in

the denominator.

36.

Using the Formula for to Find a Summation

Evaluate

Since the series is in the form it represents the sum of the first nterms

of the geometric sequence with and

Formula for

Let

= 90 Simplify. NOW TRY

=

611 - 162

- 1

S 4 = n=4,a 1 =6, r=2.

611 - 242

1 - 2

Sn= Sn

a 111 - rn 2

1 - r

a 1 = a#b^1 r= b.

a

n

i= 1

a#bi,

a

4

i= 1

3 # 2 i.

EXAMPLE 6 Sn

Evaluate. Subtract in

the denominator.

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