Is the formula $a_n = a_1\cdot r^{n-1}$ always applied to geometric sequences no matter what the case. i.e. can I rely on using this formula to solve any geometric equation or does it only apply to some?
geometric sequences formula
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sequences-and-series
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0How are you defining a geometric sequence? – 2017-02-11
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0"A geometric sequence goes from one term to the next by always multiplying (or dividing) by the same value." - http://www.purplemath.com/modules/series3.htm – 2017-02-11
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0Ok...so if $a_2=a_1\times r$ then $a_3=a_2\times r = a_1\times r^2$ and so on. – 2017-02-11
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0no a2 = a1*r^2-1 and then same concept applies to a3 with 3-1 – 2017-02-11
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0How does that differ from what I wrote? – 2017-02-11
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0the common ratio 'r' is to the power of n-1, in the example of n being 2, you have to do 2-1, which leaves you with a1*r^1 – 2017-02-12
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0Yes, which is what I wrote. Since $2-1=1$ writing $r^{2-1}$ is the same as writing $r^1$ or just $r$. I'm really not seeing your point here. – 2017-02-12
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0Sorry my mistake, I thought you wern't implying that r is the same as r^1 because you didnt really show any working out. – 2017-02-12
1 Answers
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If you work with a geometric sequence $(a_n)_n$ where $$a_n = ra_{n-1} \qquad \text{for all } n\in \mathbf N_{\geq 2}$$ you can write $$a_n= r^{n-1} a_1.$$ The formula for $a_n$ can be easily shown by induction: $$a_n = ra_{n-1} = r^2 a_{n-2} = \dotsc=r^{n-2} a_2 = r^{n-1} a_1.$$