Question

How do you determine whether the sequence 3, 12, 48, 192, … is geometric and if it is, what is the common ratio?

Answer 1

Hint:
 The sequence whose terms are obtained by the multiplication of a number to its previous term called common ratio or multiplier. Such a sequence is the geometric sequence. The generalized form of terms of a geometric sequence is $$a_n=a_0{q}^{n-1}$$, this gives the nth term where $$a_0$$ is the first term and q is the multiplier.

Complete step by step answer:
In the given question, the sequence is 3, 12, 48, 192, …
Now, let the first term,$$a_0$$ = 3. The second term is $$a_1$$=12, third term is $$a_2$$ = 48 and the fourth term is $$a_3$$ = 192.
Now let us divide the second term with first term then we get
$${\displaystyle \frac{a_1}{a_0}}$$ = $${\displaystyle \frac{12}{3}}$$ which is equal to 4 ---(1)
Now we divide the third term by second term then we get
$${\displaystyle \frac{a_2}{a_1}}$$ = $${\displaystyle \frac{48}{12}}$$ which is also equal to 4 --(2)
Now we divide the fourth term by the third term then we get
$${\displaystyle \frac{a_3}{a_2}}$$ = $${\displaystyle \frac{192}{48}}$$ which is also equal to 4 --(3)
From equations (1), (2) and (3), we can say that the terms have a common ratio equal to 4 and hence it is a geometric sequence. And the nth term of this sequence is given by $$a_n=3{(4)}^{n-1}$$.

Note:
While solving questions from a geometric sequence, one common error would be not correctly finding the value of r, the common multiplier. Sometimes sequences of fractions are confusing. You might check that the r calculated is consistently true for any two successive terms of the sequence. This helps to verify the sequence.