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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 1st term and is the common ratio, then represents a G.P.

1) If there is no change in the terms.

Sum to n terms of a G.P.

Example 1:
Find the G.P whose 4th term is 8 and 8th term is


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


512 is the 10th term.

Example 3:
Find the least value of for which the sum terms is greater than 7000.


lies between 8 and 9.
least value of is 9.

Example 4:
Find the sum to terms of the series: 0.4+0.94+0.994+......


Example 5:
Find the sum to n terms of the series: 11+103+1005+........


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?

Let be the 1st term and be the common ratio

There are infinite solutions for the equation
One such solution may be
i.e.27 is the 1st term, 8 is the 4th term and 12 is the 3rd term of a G.P.

Example 7:


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 4th , find the four numbers.

Let the last three numbers be
Since 1st & 4th numbers are the same, the four numbers can be assumed to be

The first three numbers are in G.P.

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