π Examples 1 to 7. **Write the first 3 terms of the infinite geometric series with the given characteristics**.

1. *S*=12, *r*=Β½

β Solution:

Substitute *S*=12 and *r*=Β½ into the formula for the sum of an infinite geometric series to find *a*_{1}.

Use

*r*=Β½ to find

*a*

_{2}and

*a*

_{3}.

3β Β½=3/2.

Therefore, the first three terms of the sequence are 6, 3, and 3/2.

2. *S*=-25, *r*=0.2

β Solution:

Substitute *S*=-25 and *r*=0.2 into the formula for the sum of an infinite geometric series to find *a*_{1}.

Use

*r*=0.2 to find

*a*

_{2}and

*a*

_{3}.

-4β 0.2=-β .

Therefore, the first three terms of the sequence are -20, -4, and -β .

3. *S*=-60, *r*=0.4

β Solution:

Substitute *S*=-60 and *r*=0.4 into the formula for the sum of an infinite geometric series to find *a*_{1}.

Use

*r*=0.4 to find

*a*

_{2}and

*a*

_{3}.

-14.4(0.4)=-5.76.

Therefore, the first three terms of the sequence are -36, -14.4, and -5.76.

4. *S*=β
, *a*_{1}=8/9

β Solution:

Substitute *S*=β
and *a*_{1}=8/9 into the formula for the sum of an infinite geometric series to find *r*.

Use

*r*=β to find

*a*

_{2}and

*a*

_{3}.

Therefore, the first three terms of the sequence are 8/9, -8/27, and 8/81.

5. *S*=-115, *a*_{1}=-138

β Solution:

Substitute *S*=-115 and *a*_{1}=-138 into the formula for the sum of an infinite geometric series to find *r*.

Use

*r*=-0.2 to find

*a*

_{2}and

*a*

_{3}.

27.6β (-0.2)=-5.52.

Therefore, the first three terms of the sequence are -138, 27.6, and -5.52.

6. *S*=44.8, *a*_{1}=56

β Solution:

Substitute *S*=-25 and *a*_{1}=56 into the formula for the sum of an infinite geometric series to find *r*.

Use

*r*=-0.25 to find

*a*

_{2}and

*a*

_{3}.

-14(-0.25)=3.5.

Therefore, the first three terms of the sequence are 56, -14, and 3.5.

7. *S*=-126.25, *a*_{1}=-50.5

β Solution:

Substitute *S*=-126.25 and *a*_{1}=-50.5 into the formula for the sum of an infinite geometric series to find *r*.

Use

*r*=0.6 to find

*a*

_{2}and

*a*

_{3}.

-30.3β 0.6=-18.18.

Therefore, the first three terms of the sequence are -50.5, -30.3, and -18.18.

π Ex8. The sum to infinity of a geometric series is 36 and the second term of the series is 8. Find two possible series.

β Solution:

Let the series be *a*+*ar*+*aβ
r*^{2}+ β¦

Given: *S*_{β}=36

Given:

*u*

_{2}=8.β΄

*aβ r*=8.

We now solve between (1) and (2):

*aβ r*=8 β¦ (2)

36(1-

*r*)

*r*=8 [replace

*a*with 36(1-

*r*)]

(36-36

*r*)

*r*=8

36

*r*-36

*r*

^{2}=8

-36

*r*

^{2}+36

*r*-8=0

36

*r*

^{2}-36

*r*+8=0

9

*r*

^{2}-9

*r*+2=0

(3

*r*-2)(3

*r*-1)=0

3

*r*-2=0 or 3

*r*-1=0

3

*r*=2 or 3

*r*=1

*r*=β or

*r*=β .

Put

*r*=β and

*r*=β into (1) or (2) to find the value of

*a*.

Thus, we have two series which obey the two given conditions:

(i)

*a*=12,

*r*=β , the series is 12+8+5β + β¦

(ii)

*a*=24,

*r*=β , the series is 24+8+2β + β¦