Verified Answer :

LammettHash

\(-0.1=-\dfrac15\times0.5\)
\(0.02=-\dfrac15\times(-0.1)\)
\(-0.004=-\dfrac15\times0.02\)
\(0.0008=-\dfrac15\times(-0.004)\)

This means the common ratio between the terms in the sequence is \(-\dfrac15\), so you know the recursive formula is

\(a_n=-\dfrac15a_{n-1}\)

which you can solve recursively to find an explicit formula in terms of \(a_1=0.5\).

\(a_n=-\dfrac15a_{n-1}=\left(-\dfrac15\right)^2a_{n-2}=\left(-\dfrac15\right)^3a_{n-3}=\cdots=\left(-\dfrac15\right)^{n-1}a_1\)

So the explicit formula is

\(a_n=\dfrac12\left(-\dfrac15\right)^{n-1}=0.5\times(-0.2)^{n-1}\)

apologiabiology

geometric sequence
an=a1(r)^(n-1)
r=common ratio
a1=first term
an=nth term

the common ratio is found by dividing a term by the term before it

-0.1/0.5=-1/5

the common ratio is -1/5
the 1th term is 0.05

the formula is
an=0.5(-1/5)^(n-1) or
an=-2.5(-1/5)^n

Related Questions