Question

What is the recursive formula for the geometric sequence with this explicit formula?

Answer 1

`\large\huge\boxed{\left\{\begin{array}{ccc}a_1=9\\a_n=a_(n-1)\cdot\left(-(1)/(3)\right)\end{array}\right}`

Explanation:

`a_n=9\cdot\left(-(1)/(3)\right)^(n-1)\\\\\text{Calculate}\ a_1.\ \text{Put n = 1 to the explicit formula of the geometric sequence:}\\\\a_1=9\cdot\left(-(1)/(3)\right)^(1-1)=9\cdot\left(-\dfraC{1}{3}\right)^0=9\cdot1=9\\\\\text{Calculate the common ratio:}\\\\r=(a_(n+1))/(a_n)\\\\a_(n+1)=9\cdot\left(-(1)/(3)\right)^(n+1-1)=9\cdot\left(-(1)/(3)\right)^n`

`r=(9\!\!\!\!\diagup^1\cdot\left(-(1)/(3)\right)^n)/(9\!\!\!\!\diagup_1\cdot\left(-(1)/(3)\right)^(n-1))\qquad\text{use}\ (a^m)/(a^n)=a^(m-n)\\\\r=\left(-(1)/(3)\right)^(n-(n-1))=\left(-(1)/(3)\right)^(n-n-(-1))=\left(-(1)/(3)\right)^1=-(1)/(3)\\\\a_n=a_(n-1)\cdot\left(-(1)/(3)\right)`