# 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

π Question 1. How many terms of the geometric sequence 3, 32, 33, β¦ are needed to give the sum 120?

3+32+33+34β¦120
3+9+27+81=120

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

π 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
β 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.

Therefore, the correct answer is (A).

π Q3. How many terms of the geometric sequence 3, 3/2, ΒΎ, β¦ are needed to give the sum 3069/512?
β Solution: Let n be the number of terms needed. Given that a=3, r=Β½ and Sn=3069/512

which gives n=10.

π 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.
β Solution:
Let the sum of n terms of the geometric sequence be 315.
It is known that,

It is given that the first term a is 5 and common ratio r is 2.

β΄ Last term of the geometric sequence = 6th term =aβr(6-1)=5β25=5β32=160 Thus, the last term of the geometric sequence is 160.

π Activity 1.
1. Determine 3+6+12+24+β¦ to 10 terms
2. If 2+6+18+β¦=728, determine the value of n.
β Solutions:

π Q4. How many terms in the geometric sequence 1, 1.1, 1.21, 1.331, β¦ will be needed so that the sum of the first n terms is greater than 20?
β 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

If we now take logarithms of both sides, we get
n lnβ‘1.1>lnβ‘3

and as lnβ‘1.1>0 we obtain
n>lnβ‘3βlnβ‘1.1 =11.5267β¦

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

π 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
β Solution:
General sum Formula

Therefore, D is the correct answer.

π Ex2. If Sn=61/40, β+Β½+β+β―, find n.
β Solution:
Find the common ratio.

Β½Γ·β=β
βΓ·Β½=β

Substitute Sn=61/40, a1=β, and r=β into the formula for the nth partial sum of an infinite geometric series.

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

π Ex3. Find n for 4.1+8.2+16.4+β― if Sn=61.5.
β 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.

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