Question

Write the equation of the arithmetic and geometricsequences given 15, 5, ...a. arithmeticb. geometric

Answer 1

The rate in a arithmetic sequence can be determined by the subtraction of two subsequent terms:

$a_2-a_1=r$

So, using the second term equal to 5 and the first one equal to 15, we have:

$\begin{gathered} 5-15=r \\ r=-10 \end{gathered}$

Now, using the formula for the nth term of a arithmetic sequence, we have:

$\begin{gathered} a_n=a_1+(n-1)\cdot r \\ a_n=15-10(n-1) \end{gathered}$

Now, for the geometric sequence, the rate is given by the division of two subsequent terms:

$\frac{a_2}{a_1}=q$

So we have that:

$\begin{gathered} \frac{5}{15}=q \\ q=\frac{1}{3} \end{gathered}$

The nth term of a geometric sequence is given by:

$\begin{gathered} a_n=a_1\cdot q^{n-1} \\ a_n=15\cdot(\frac{1}{3})^{n-1} \end{gathered}$