# Geometric Progression (G.P)

Let us consider the following sequences:

(3) 1, 0.1, 0.01, 0.001,.....

These types of sequences, where there is a constant ratio (called common ratio) between two successive terms like second & first; third and second and so on, are called geometric sequence or geometric progression.

If is the 1

^{st}term and is the common ratio, then represents a G.P.

**Note:**

1) If there is no change in the terms.

# Sum to n terms of a G.P.

**Example 1:**

Find the G.P whose 4

^{th}term is 8 and 8

^{th}term is

**Solution:**

**Example 2:**

Which term of the G.P 1, 2, 4, 8, ...... is 512?

**Solution:**

âˆ´ 512 is the 10

^{th}term.

**Example 3:**

Find the least value of for which the sum terms is greater than 7000.

**Solution:**

âˆ´ lies between 8 and 9.

âˆ´ least value of is 9.

**Example 4:**

Find the sum to terms of the series: 0.4+0.94+0.994+......

**Solution:**

**Example 5:**

Find the sum to n terms of the series: 11+103+1005+........

**Solution:**

**Example 6:**

Does there exist a G.P containing 27, 8, 12 as three of its terms. If it exists, how many such progressions are possible?

**Solution:**

Let be the 1

^{st}term and be the common ratio

There are infinite solutions for the equation

One such solution may be

i.e.27 is the 1

^{st}term, 8 is the 4

^{th}term and 12 is the 3

^{rd}term of a G.P.

**Example 7:**

**Solution:**

**Example 8:**

In a set of four numbers, the first three are in G.P and the last three are in A.P with a common difference 6. If the first number is the same as the 4

^{th}, find the four numbers.

**Solution:**

Let the last three numbers be

Since 1

^{st}& 4

^{th}numbers are the same, the four numbers can be assumed to be

The first three numbers are in G.P.