Are two geometric sequences with the first $a_1$ and the same common ratio identical?

Question

If I have two geometric sequences, and they both have the same first term and the same common ratio, can I say they are the same?

For example:

$a_n = 3*0.5^{n-1}$

$b_n = \frac{3}{2^{n-1}}$

Answer 1

Yes, because you what you have written is the same geometric series written in decimal and fractional form.

Note that because $0.5=\frac{1}2$ we start with your first geometric sequence:

• $a_n=3(0.5)^{n-1}$

Next we can replace $0.5$ with $\frac{1}2$:

• $a_n=3(\frac{1}2)^{n-1}=3(\frac{1}{2^{n-1}})$

Which simplifies to our final answer

• $a_n=\frac{3}{2^{n-1}}$

So we see that $a_n=b_n$ (as required).