Exercise · ZW-RECURRENCE-RELATION-004 · ★☆☆ Basic
If \(a_0=3\) and \(a_n=2a_{n-1}\), which is a formula for \(a_n\)?
Choose one option.
Show answer
- C. \(a_n=3\cdot2^n\)
Explanation
Each step multiplies by 2, so \(a_n=3\cdot2^n\); check by induction. No other option gives \(a_1=6\) and \(a_2=12\).
Review the concept: Recurrence relation
A recurrence relation is an equation that expresses each term \(a_n\) of a sequence in terms of the previous \(k\) terms; with initial values, it determines the whole sequence.
\[a_n=f(a_{n-1},a_{n-2},\dots,a_{n-k})\]
More practice problems: recurrence relation
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- ZW-RECURRENCE-RELATION-003 Let a_0=1 and a_n=a_n-1+2n. What is a_5?
- ZW-RECURRENCE-RELATION-005 In the Tower of Hanoi, the minimum number of moves satisfies T_n=2T_n-1+1 with T_0=0. What is T_6?
- ZW-RECURRENCE-RELATION-006 How many initial values are needed to determine a sequence with a_n=a_n-1+a_n-2?
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- ZW-RECURRENCE-RELATION-008 The sequence with a_0=1,\ a_1=3 and a_n=2a_n-1-a_n-2 satisfies which formula?
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