HOW TO DETERMINE IF THE GIVEN SEQUENCE IS GEOMETRIC SEQUENCE

How to Determine if the Given Sequence is Geometric Sequence ?

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.

If the common ratios are equal, then the given sequence is geometric sequence.

r = t_{2}/t_{1} (or) t_{3}/t_{2 } (or) t_{4}/t_{3}

How to Determine if the Given Sequence is Geometric Sequence - Questions

Question 1 :

Which of the following sequences are in G.P.?

(i) 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

Since the common ratios are same, it is geometric sequence.

(ii) 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

Since the common ratios are not same, it is not geometric sequence.

(iii) 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

Since the common ratios are same, it is geometric sequence.

(iv) 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

Since the common ratios are same, it is geometric sequence.

(v) 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

Since the common ratios are same, it is geometric sequence.

(vi) 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

Since the common ratios are same, it is geometric sequence.

(vii) 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

Since the common ratios are same, it is geometric sequence.

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