A recursive function \(ℎ\), is defined as follows: \(\begin{array} {} h(m) &…
2017
A recursive function \(ℎ\), is defined as follows:
\(\begin{array} {} h(m) & =k, \text{if } m=0 \\ &=1, \text{if } m=1 \\ &= 2 h(m-1)+4h(m-2), \text{if } m \geq 2 \end{array}\)
If the value of \(ℎ(4)\) is \(88\) then the value of \(𝑘\) is:
- A.
0
- B.
1
- C.
2
- D.
- 1
Attempted by 53 students.
Show answer & explanation
Correct answer: C
Key idea: use the recurrence to express h(2), h(3), and h(4) in terms of k, then solve for k using h(4)=88.
Given: h(0) = k and h(1) = 1.
Compute h(2): h(2) = 2·h(1) + 4·h(0) = 2·1 + 4·k = 2 + 4k.
Compute h(3): h(3) = 2·h(2) + 4·h(1) = 2(2 + 4k) + 4·1 = 8 + 8k.
Compute h(4): h(4) = 2·h(3) + 4·h(2) = 2(8 + 8k) + 4(2 + 4k) = 24 + 32k.
Set h(4) = 88 and solve: 24 + 32k = 88 ⇒ 32k = 64 ⇒ k = 2.
Answer: k = 2