Question

the table shows the number of visitors to a web page over a period of several months. write an explicit formula and a recursive formula to model the data.
month \t1 \t2 \t3 \t4 \t5
visitors \t15 \t30 \t60 \t120 \t240

the explicit formula is $a_n = 15cdot 2^{n - 1}$.

the recursive formula is $a_1=square$ and $a_n=square$.

Explanation:

Step1: Find \(a_1\)

The first term (when \(n = 1\)) of the sequence is the number of visitors in month 1, which is 15. So \(a_1=15\).

Step2: Determine the recursive relation

Observe the sequence: \(15, 30, 60, 120, 240, \dots\). Notice that each term is double the previous term. For example, \(30 = 2\times15\), \(60 = 2\times30\), \(120 = 2\times60\), etc. So the recursive formula for a geometric sequence (where each term is a multiple of the previous term) is \(a_n = 2\times a_{n - 1}\) for \(n>1\).

Answer:

The recursive formula is \(a_1 = \boldsymbol{15}\) and \(a_n=\boldsymbol{2a_{n - 1}}\) (for \(n\geq2\)).