HOW TO DETERMINE IF THE GIVEN SEQUENCE IS GEOMETRIC SEQUENCE
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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.
To determine whether a sequence is geometric or not, we have to find the ratio between the first term and second term, second term and third term, third term and fourth term.
That is,
t2/t1, t3/t2 and t4/t3
If all the above ratios are equal, then the given sequence is geometric sequence.
Practice Problems
Problem 1 :
Check whether the following sequence is geometric or not.
3, 9, 27, 81, ..........
Solution :
First term (a) = 3, common ratio
|
r = t2/t1 = 9/3 r = 3 |
r = t3/t2 = 27/9 r = 3 |
r = t2/t1 = 81/27 r = 3 |
Because the ratios are same, it is geometric sequence.
Problem 2 :
Check whether the following sequence is geometric or not.
4, 44, 444, 4444, ..........
Solution :
First term (a) = 3, common ratio
|
r = t2/t1 = 44/4 r = 11 |
r = t3/t2 = 444/44 r = 111/11 |
r = t2/t1 = 4444/444 r = 1111/111 |
Because the ratios are same, it is geometric sequence.
Problem 3 :
Check whether the following sequence is geometric or not.
0.5, 0.05, 0.005, ..........
Solution :
First term (a) = 0.5, common ratio
|
r = t2/t1 = 0.05/0.5 r = 0.1 |
r = t2/t1 = 0.005/0.05 r = 0.1 |
Because the ratios are same, it is geometric sequence.
Problem 4 :
Check whether the following sequence is geometric or not.
1/3, 1/6, 1/12, ..........
Solution :
First term (a) = 1/3, common ratio
|
r = t2/t1 = (1/6)/(1/3) r = 1/2 |
r = t2/t1 = (1/12)/(1/6) r = 1/2 |
Because the ratios are same, it is geometric sequence.
Problem 5 :
Check whether the following sequence is geometric or not.
1, −5, 25, −125, ..........
Solution :
First term (a) = 1/3, common ratio
|
r = t2/t1 = -5/1 r = -5 |
r = t2/t1 = 25/(-5) r = -5 |
Because the ratios are same, it is geometric sequence.
Problem 6 :
Check whether the following sequence is geometric or not.
120, 60, 30, 18, ..........
Solution :
First term (a) = 120, common ratio
|
r = t2/t1 = 60/120 r = 1/2 |
r = t2/t1 = 30/60 r = 1/2 |
Because the ratios are same, it is geometric sequence.
Problem 7 :
Check whether the following sequence is geometric or not.
16, 4, 1, 1/4, ..........
Solution :
First term (a) = 16, common ratio
|
r = t2/t1 = 4/16 r = 1/4 |
r = t2/t1 r = 1/4 |
Because the ratios are same, it is geometric sequence.
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