Write the First 3 terms of each infinite geometric series, Given each Sum

💎 Examples 1 to 7. Write the first 3 terms of the infinite geometric series with the given characteristics.
1. S=12, r
✍ Solution:
Substitute S=12 and r=½ into the formula for the sum of an infinite geometric series to find a1.

get first term e

Use r=½ to find a2 and a3.
6⋅½=3
3⋅½=3/2.

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

2. S=-25, r=0.2
✍ Solution:
Substitute S=-25 and r=0.2 into the formula for the sum of an infinite geometric series to find a1.

get first term f

Use r=0.2 to find a2 and a3.
-20⋅0.2=-4
-4⋅0.2=-⅘.

Therefore, the first three terms of the sequence are -20, -4, and -⅘.

3. S=-60, r=0.4
✍ Solution:
Substitute S=-60 and r=0.4 into the formula for the sum of an infinite geometric series to find a1.

get first term g

Use r=0.4 to find a2 and a3.
-36(0.4)=-14.4
-14.4(0.4)=-5.76.

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

4. S=⅔, a1=8/9
✍ Solution:
Substitute S=⅔ and a1=8/9 into the formula for the sum of an infinite geometric series to find r.

get first term h

Use r=⅓ to find a2 and a3.
get first term h2

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

5. S=-115, a1=-138
✍ Solution:
Substitute S=-115 and a1=-138 into the formula for the sum of an infinite geometric series to find r.

first three terms a

Use r=-0.2 to find a2 and a3.
-138⋅(-0.2)=27.6
27.6⋅(-0.2)=-5.52.

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

6. S=44.8, a1=56
✍ Solution:
Substitute S=-25 and a1=56 into the formula for the sum of an infinite geometric series to find r.

first three terms b

Use r=-0.25 to find a2 and a3.
56(-0.25)=-14
-14(-0.25)=3.5.

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

7. S=-126.25, a1=-50.5
✍ Solution:
Substitute S=-126.25 and a1=-50.5 into the formula for the sum of an infinite geometric series to find r.

common ratio from sum to infinity a

Use r=0.6 to find a2 and a3.
-50.5⋅0.6=-30.3
-30.3⋅0.6=-18.18.

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

📌 Ex8. The sum to infinity of a geometric series is 36 and the second term of the series is 8. Find two possible series.
✍ Solution:
Let the series be a+ar+a⋅r2+ …
Given: S=36

get first term i

Given: u2=8.∴a⋅r=8.
We now solve between (1) and (2):
a⋅r=8 … (2)
36(1-r)r=8 [replace a with 36(1-r)]
(36-36r)r=8
36r-36r2=8
-36r2+36r-8=0
36r2-36r+8=0
9r2-9r+2=0
(3r-2)(3r-1)=0
3r-2=0 or 3r-1=0
3r=2 or 3r=1
r=⅔ or r=⅓.

Put r=⅔ and r=⅓ into (1) or (2) to find the value of a.
get first term j

Thus, we have two series which obey the two given conditions:
(i) a=12, r=⅔, the series is 12+8+5⅓+ …
(ii) a=24, r=⅓, the series is 24+8+2⅔+ …

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