Given $S= {{1,2,3,4,5,6,7,8}}$ ; How many infinite sequences can be formed from $S$ elements such that each sequence is both Geometric and Arithmetic sequence ?
The answer is $8$ in the book ; but i don't understand how to get this answer?
Any help will be appreciated
Thank you for your help
Hint: If $(a_n)$ is an arithmetic sequence, then we have $a_n=a_0+nd.$ If $(a_n)$ is also geometric, we have $a_n=r^da_0$. Looking at $a_1$, $$ a_0+d=ra_0 $$ So $d=(r-1)a_0$. Looking at $a_2$, $a_0+2d=r^2a_0$. Substituting for $d$ gives $a_0+2(r-1)a_0=r^2a_0$, which is a quadratic equation that you can solve for $r$.