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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.

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 4th term is 8 and 8th term is

Solution:


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

Solution:

512 is the 10th term.

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

Solution:

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+......

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 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:

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 4th , find the four numbers.

Solution:
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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