Geometric Progression - find ith and r

Question

Hi great if you could please help me solve this:

Given: $$a_0=12$$

$$a_5=24$$

$$a_{10} =48$$

thus ratio $$a_0:a_5 = 2$$

$n =$ number of terms in scale $= 5 (a_0, a_1, a_2, a_3, a_4) $

$a_0 =$ first term $= 12$

What is the equation to find $i$-th value $a_i$ , and the common ratio.

Thanks Jean-Lou

Answer 1

$i$-th term of the GP can be given by:$$a_i=a_0\times{r}^i$$

now $r$ can be determined using the above formula:$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ a_5=a_0\times r^5$$$$\ \ \ \ \ \ \ \ \Rightarrow 24=12\times r^5$$$$\ \ \ \ \ \ \ \ \Rightarrow r^5=\frac{24}{12}=2$$$$\Rightarrow r=2^{1/5} $$

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Note that this formula for $i$-th term is not always going to be valid. In this case it is valid because the initial term of your GP is called $a_0$. Sometimes, the initial term of a GP is called $a_1$, in that case the formula would be:$$a_i=a_1\times r^{i-1}$$

Answer 2

Hints: (HUGE Hints)

1) If $\frac {a_{k+1}}{a_{k}} = r$ then what is

$\frac {a_5}{a_4}\frac {a_4}{a_3}\frac {a_3}{a_2}\frac {a_2}{a_1}\frac{a_1}{a_0}$? And if $\frac {a_5}{a_0} = 2$ then what is $r$?

2) And if $a_{i} = r*{a_{i-1}} = r*(r*a_{i-2}) = r*(r*(r*a_{i-3})) = ...$

3) And if $a_i = f(i)$ be the formula in hint 2) then $a_5 = 2a_0$ then $f(5) = 2\times f(0)$. Can that tell you what $r$ is?

....

Do any two of the three hints. Do all three if you need to.