Exercise · ZW-STRONG-INDUCTION-006 · ★★☆ Intermediate
Let \(a_1=1\), \(a_2=2\) and \(a_n=a_{n-1}+a_{n-2}\). What is \(a_8\)?
Choose one option.
Show answer
- D. \(34\)
Explanation
The terms are \(1,2,3,5,8,13,21,34\). The value \(21\) is \(a_7\).
Review the concept: Strong induction
Strong induction proves \(P(n)\) for every \(n\ge n_0\): after checking the base case \(P(n_0)\), one proves \(P(k+1)\) assuming \(P(j)\) for all \(j\) with \(n_0\le j\le k\).
\[\big[P(n_0)\wedge\forall k\ge n_0\,(P(n_0)\wedge\cdots\wedge P(k)\Rightarrow P(k+1))\big]\Rightarrow\forall n\ge n_0\,P(n)\]
More practice problems: strong induction
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- ZW-STRONG-INDUCTION-005 Compared with ordinary mathematical induction, strong induction:
- ZW-STRONG-INDUCTION-007 With stamps of 4 and 5 cents, what is the least n_0 such that every amount n≥ n_0 can be made?
- ZW-STRONG-INDUCTION-008 Let a_0=0, a_1=1 and a_n=3a_n-1-2a_n-2. It is proved by strong induction that a_n=2^n-1. What is a_…
- ZW-STRONG-INDUCTION-009 It is proved that every amount n≥12 can be made with stamps of 4 and 5, going from n-4 to n. How ma…