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

Geometric Sequence (G.S.)
Let us consider the following sequences:

(i) 2, 4, 8, 16, …, (ii) 1/9, (-1)/27, 1/81, (-1)/243, … (iii) .01, .0001, .000001, …
In each of these sequences, how their terms progress? We note that each term, except the first progresses in a definite order.

In (i), we have a1=2, a2/a1 =2, a3/a2 =2, a4/a3 =2 and so on.

In (ii), we observe, a1=1/9, a2/a1 =-⅓, a3/a2 =-⅓, a4/a3 =-⅓ and so on.

Similarly, state how do the terms in (iii) progress? It is observed that in each case, every term except the first term bears a constant ratio to the term immediately preceding it. In (i), this constant ratio is 2; in (ii), it is -⅓ and in (iii), the constant ratio is 0.01.
Such sequences are called geometric sequence or geometric progression abbreviated as GS or GP.
A sequence a1, a2, a3, …, an is called geometric sequence, if each term is non-zero and a(k+1)/ak =r (constant), for k≥1.

By letting a1=a, we obtain a geometric progression, a, a⋅r, a⋅r2, a⋅r3, …, where a is called the first term and r is called the common ratio of the geometric sequence. Common ratio in geometric progression (i), (ii) and (iii) above are 2, -⅓ and 0.01, respectively.
As in case of arithmetic progression, the problem of finding the nth term or sum of n terms of a geometric progression containing a large number of terms would be difficult without the use of the formulae which we shall develop in the next Section. We shall use the following notations with these formulae:

a= the first term, r= the common ratio, = the last term, n= the numbers of terms, Sn= the sum of first n terms and so an= the last term.

📌 Example 1. If S6=196.875, a1=100, r=0.5, find a6.
✍ Solution:
Substitute S6=196.875, a1=100, and r=0.5 into the formula for the nth partial sum of an infinite geometric series.

get last term a

📌 Ex2. If r=-0.4, S5=144.32, and a1=200, find a5.
✍ Solution:
Substitute S5=144.32, a1=200, and r=-0.4 into the formula for the nth partial sum of an infinite geometric series.

get last term b

📌 Ex3. If 15-18+21.6-…, Sn=23.784, find an.
✍ Solution:
Find the common ratio.

-18÷15=-1.2
21.6÷-18=-1.2

Substitute Sn=23.784, a1=15, and r=-1.2 into the formula for the nth partial sum of an infinite geometric series.
get last term c

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