Question

A recursive rule for a geometric sequence is a1=2; an= 4/9 an-1

What is the explicit rule for this sequence?

Answer 1

A recursive rule for a geometric sequence:

`a_1\\\\a_n=r\cdot a_(n-1)`

---------------------------------------------------

`a_1=2;\ a_n=(4)/(9)a_(n-1)\to r=(4)/(9)`

---------------------------------------------------

The exciplit rule:

`a_n=a_1r^(n-1)`

Substitute:

`a_n=2\left((4)/(9)\right)^(n-1)=2\cdot\left((4)/(9)\right)^n\cdot\left((4)/(9)\right)^(-1)=2\cdot\left((4)/(9)\right)^n\cdot(9)/(4)\\\\\boxed{a_n=(9)/(2)\cdot\left((4)/(9)\right)^n}`