How do you use the geometric sequence formula to find the nth term?

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How do you use the geometric sequence formula to find the nth term?

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Answer 1 · 2015-01-13T13:11:04

A geometric sequence is always of the form
$t_{n} = t_{n-1} \cdot r$ $t_n=t_"n-1"*r$

Every next term is $r$ $r$ times as large as the one before.

So starting with $t_{0}$ $t_0$ (the "start term") we get:
$t_{1} = r \cdot t_{0}$ $t_1=r*t_0$
$t_{2} = r \cdot t_{1} = r \cdot r \cdot t_{0} = r^{2} \cdot t_{0}$ $t_2=r*t_1=r*r*t_0=r^2*t_0$
......
$t_{n} = r^{n} \cdot t_{0}$ $t_n=r^n*t_0$

Answer:
$t_{n} = r^{n} \cdot t_{0}$ $t_n=r^n*t_0$
$t_{0}$ $t_0$ being the start term, $r$ $r$ being the ratio

Extra:
If $r > 1$ $r>1$ then the sequence is said to be increasing
if $r = 1$ $r=1$ then all numbers in the sequence are the same
If $r < 1$ $r<1$ then the sequence is said to be decreasing ,
and a total sum may be calculated for an infinite sequence:
sum $\sum = \frac{t_{0}}{1 - r}$ $sum=t_0/(1-r)$

Example :
The sequence $1,1/2,1/4,1/8...$
Here the $t_0=1$ and the ratio $r=1/2$
Total sum of this infinite sequence:
$sum=t_0/(1-r)=1/(1-1/2)=2$

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How do you use the geometric sequence formula to find the nth term?

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