How to check if the given sequence is geometric or not ?
To find the common ratio, we use the formula
r = a_{2}/a_{1 }(or) r = a_{3}/a_{2 }
What is geometric progression ?
A Geometric Progression is a sequence in which each term is obtained by multiplying a fixed non-zero number to the preceding term except the first term.
The fixed number is called common ratio. The common ratio is usually denoted by r.
Example 1 :
Find out which of the following sequences are geometric sequences . For those geometric sequences, find the common ratio.
1/2, 1/3, 2/9, 4/47,...........
Solution :
r = t_{2}/t_{1} r = (1/3) / (1/2) r = 2/3 ----(1) |
r = t_{3}/t_{2} r = (2/9) / (1/3) r = 2/3 ----(2) |
Since the common ratios are same, the given sequence is a geometric sequence. The required common ratio is 2/3.
(ii) 12, 1, 1/12, ............
Solution :
r = t_{2}/t_{1} r = 1/12 ----(1) |
r = t_{3}/t_{2} r = (1/12) / 1 r = 1/12 ----(2) |
Since the common ratios are same, the given sequence is a geometric sequence. The required common ratio is 1/12.
(iii) √2, 1/√2, 1/2√2,...........
Solution :
r = t_{2}/t_{1} r = (1/√2)/√2 r = 1/2 ----(1) |
r = t_{3}/t_{2} r = (1/2√2) / (1/√2) r = 1/2 ----(2) |
Since the common ratios are same, the given sequence is geometric progression.
The required common ratio is 1/2.
(iv) 0.004, 0.02, 1, ..........
Solution :
t_{1} = 0.004, t_{2} = 0.02 and t_{3} = 1
r = t_{2}/t_{1} r = 0.02 / 0.004 r = 20 / 4 r = 5 ----(1) |
r = t_{3}/t_{2} r = 1 / 0.02 r = 100/2 r = 50 ----(2) |
Since the common ratios are not same, the given sequence is not geometric progression.
After having gone through the stuff given above, we hope that the students would have understood how to check if the given sequence is geometric or not.
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