Get First Term, Given the Sum of a Finite Geometric Series

💎 Finding Specific Terms from the Sum of the First n Terms of a Geometric Sequence

You can use the formula for the sum of a geometric series to help find a particular term of the series.
📌 Example 1: Find the First Term of a Series
Find a1 in a geometric series for which S8=39,360 and r=3.
✍ Solution:


The first term of the series is 12.

📌 Example 2: Determining the First Term of a Geometric Series
The sum of the first 10 terms of a geometric series is -29,524.
The common ratio is -3. Determine the 1st term.
✍ Solution:
Suppose the geometric series has 1st term, t1, and common ratio, r.
Use:


Substitute: n=10, Sn=-29,524, r=-3

The 1st term is 2.

📌 Ex3. Find a1 if S12=1,365 and r=2.
✍ Solution:
Substitute S12=1, 365, n=12, and r=2 into the formula for the sum of a finite geometric series.

📌 Ex4. Find a1 if Sn=468, an=375, and r=5.
✍ Solution:
Substitute Sn=468, an=375, and r=5 into the formula for the nth partial sum of an infinite geometric series.


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