**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 *a*_{1}=2, *a*_{2}/*a*_{1} =2, *a*_{3}/*a*_{2} =2, *a*_{4}/*a*_{3} =2 and so on.

In (ii), we observe, *a*_{1}=1/9, *a*_{2}/*a*_{1} =-⅓, *a*_{3}/*a*_{2} =-⅓, *a*_{4}/*a*_{3} =-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 *a*_{1}, *a*_{2}, *a*_{3}, …, *a _{n}* is called

**geometric sequence**, if each term is non-zero and

*a*

_{(k+1)}/

*a*=

_{k}*r*(constant), for

*k*≥1.

By letting *a*_{1}=a, we obtain a geometric progression, *a*, *a⋅r*, *a⋅r*^{2}, *a⋅r*^{3}, …, 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 *n*th 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, *ℓ*= the last term, *n*= the numbers of terms, *S _{n}*= the sum of first

*n*terms and so

*a*= the

_{n}**last term**.

📌 Example 1. If *S*_{6}=196.875, *a*_{1}=100, *r*=0.5, find *a*_{6}.

✍ Solution:

Substitute *S*_{6}=196.875, *a*_{1}=100, and *r*=0.5 into the formula for the *n*th partial sum of an infinite geometric series.

📌 Ex2. If *r*=-0.4, *S*_{5}=144.32, and *a*_{1}=200, find *a*_{5}.

✍ Solution:

Substitute *S*_{5}=144.32, *a*_{1}=200, and *r*=-0.4 into the formula for the *n*th partial sum of an infinite geometric series.

📌 Ex3. If 15-18+21.6-…, *S _{n}*=23.784, find

*a*.

_{n}✍ Solution:

Find the common ratio.

21.6÷-18=-1.2

Substitute

*S*=23.784,

_{n}*a*

_{1}=15, and

*r*=-1.2 into the formula for the

*n*th partial sum of an infinite geometric series.

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