Question

which recursive formula defines the sequence: 5, 15, 45, 135, ... for n > 1?
f(1) = 5, f(n) = f(n - 1) - 3
f(1) = 5, f(n) = f(n - 1) · 3
f(1) = 5, f(n) = f(n - 1) + 10
f(1) = 5, f(n) = f(n - 1) · 5

Explanation:

Step1: Analyze the sequence pattern

The sequence is \(5, 15, 45, 135, \dots\). Let's check the relationship between consecutive terms. \(15\div5 = 3\), \(45\div15 = 3\), \(135\div45 = 3\). So each term is 3 times the previous term.

Step2: Evaluate each option

• For the first option: \(f(n)=f(n - 1)-3\). If \(f(1) = 5\), then \(f(2)=5 - 3=2\), which is not 15. So this is wrong.
• For the second option: \(f(1) = 5\), \(f(n)=f(n - 1)\cdot3\). \(f(2)=5\times3 = 15\), \(f(3)=15\times3 = 45\), \(f(4)=45\times3 = 135\), which matches the sequence.
• For the third option: \(f(n)=f(n - 1)+10\). \(f(2)=5 + 10 = 15\), \(f(3)=15+10 = 25

eq45\). So this is wrong.

• For the fourth option (partially visible, but from the pattern we saw, the multiplier is 3, not 5 as the visible part suggests, so it's wrong).

Answer:

B. \(f(1) = 5, f(n)=f(n - 1)\cdot3\)