Question

what is the explicit rule for the sequence: 2, 4, 8, 16... f(n)=2^{n - 1} f(n)=2(n) f(n)=2(2)^{n - 1} f(n)=2+(2)^{n - 1}

Explanation:

Step1: Identify sequence type

The sequence 2, 4, 8, 16... is geometric, with first term $a_1=2$ and common ratio $r=2$.

Step2: Recall geometric sequence formula

The explicit rule for a geometric sequence is $f(n)=a_1(r)^{n-1}$.

Step3: Substitute values into formula

Substitute $a_1=2$ and $r=2$: $f(n)=2(2)^{n-1}$.

Step4: Verify with sequence terms

For $n=1$: $2(2)^{1-1}=2(1)=2$; $n=2$: $2(2)^{2-1}=4$; $n=3$: $2(2)^{3-1}=8$, which matches the sequence.

Answer:

C. $f(n)=2(2)^{n-1}$