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,
t_{2}/t_{1, } t_{3}/t_{2 }and t_{4}/t_{3}
If all the above ratios are equal, then the given sequence is geometric sequence.
Problem 1 :
Check whether the following sequence is geometric or not.
3, 9, 27, 81, ..........
Solution :
First term (a) = 3, common ratio
r = t_{2}/t_{1} = 9/3 r = 3 |
r = t_{3}/t_{2} = 27/9 r = 3 |
r = t_{2}/t_{1} = 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 = t_{2}/t_{1} = 44/4 r = 11 |
r = t_{3}/t_{2} = 444/44 r = 111/11 |
r = t_{2}/t_{1} = 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 = t_{2}/t_{1} = 0.05/0.5 r = 0.1 |
r = t_{2}/t_{1} = 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 = t_{2}/t_{1} = (1/6)/(1/3) r = 1/2 |
r = t_{2}/t_{1} = (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 = t_{2}/t_{1} = -5/1 r = -5 |
r = t_{2}/t_{1} = 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 = t_{2}/t_{1} = 60/120 r = 1/2 |
r = t_{2}/t_{1} = 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 = t_{2}/t_{1} = 4/16 r = 1/4 |
r = t_{2}/t_{1} r = 1/4 |
Because the ratios are same, it is geometric sequence.
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