Calculating the number of Terms of the nth partial sum of a Geometric Series

Formulae for the nth partial sum of an infinite geometric series are

formulae for geometric series

📌 Question 1. How many terms of the geometric sequence 3, 32, 33, … are needed to give the sum 120?
Answer

3+32+33+34…120
3+9+27+81=120

Thus, four terms of the given geometric sequence are required to obtain the sum as 120.

📌 Q2. The first term of a geometric series is -1, and the common ratio is -3. How many terms are in the series if its sum is 182?
(A) 6 (B)7 (C) 8 (D) 9
✍ Solution:
The first term in the series is -1, the common ratio is -3, and the sum is 182. Use the formula for the sum of a finite geometric series to find the number of terms n.

how many terms

Therefore, the correct answer is (A).

📌 Q3. How many terms of the geometric sequence 3, 3/2, ¾, … are needed to give the sum 3069/512?
✍ Solution: Let n be the number of terms needed. Given that a=3, r=½ and Sn=3069/512

how many terms a

which gives n=10.

📌 Example 1. The sum of some terms of a geometric sequence is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.
✍ Solution:
Let the sum of n terms of the geometric sequence be 315.
It is known that,

sum bd

It is given that the first term a is 5 and common ratio r is 2.
how many terms b

∴ Last term of the geometric sequence = 6th term =a⋅r(6-1)=5⋅25=5⋅32=160 Thus, the last term of the geometric sequence is 160.

📌 Activity 1.
1. Determine 3+6+12+24+… to 10 terms
2. If 2+6+18+…=728, determine the value of n.
✍ Solutions:

the value of n

📌 Q4. How many terms in the geometric sequence 1, 1.1, 1.21, 1.331, … will be needed so that the sum of the first n terms is greater than 20?
✍ Solution:
The sequence is a geometric sequence with a=1 and r=1.1. We want to find the smallest value of n such that Sn>20. Now

how many terms c

If we now take logarithms of both sides, we get
n ln⁡1.1>ln⁡3

and as ln⁡1.1>0 we obtain
n>ln⁡3⁄ln⁡1.1 =11.5267…

and therefore the smallest whole number value of n is 12.

📌 Q5. The first term of a geometric series is 5, and the common ratio is -2. How many terms are in the series if its sum is -6825?
A 5 B 9 C 10 D 12
✍ Solution:
General sum Formula

how many terms d

Therefore, D is the correct answer.
Answer: D

📌 Ex2. If Sn=61/40, ⅝+½+⅖+⋯, find n.
✍ Solution:
Find the common ratio.

½÷⅝=⅘
⅖÷½=⅘

Substitute Sn=61/40, a1=⅝, and r=⅘ into the formula for the nth partial sum of an infinite geometric series.
find n

Because an=⅖ and the third term of the sequence is ⅖, n=3.

📌 Ex3. Find n for 4.1+8.2+16.4+⋯ if Sn=61.5.
✍ Solution:
The common ratio is 2.
Substitute Sn=61.5, a1=4.1, and r=2 into the formula for the nth partial sum of an infinite geometric series.

nth partial sum a

Substitute an=32.8, a1=4.1, and r=2 into the formula for the nth term of a geometric sequence to find n.
find n a

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