Solution: 2015 Winter Final - 8
Author: Michiel SmidQuestion
Consider the following recursive function:
- f(0) = $7$,
- f(n) = $f(n - 1) + 6n - 3\; \ \text{for all}$ $\mathrm{integers}\ n \geq 1$.
(a)
For all $n \geq 0$: $f(n) = 3n^2 + 7$
(b)
None of the above.
(c)
For all $n \geq 0$: $f(n) = 2n^2 + 7$
(d)
For all $n \geq 0$: $f(n) = 4n^2 + 7$
Solution
Calculate $ f(1) $
$ f(1) = f(0) + 6 \cdot 1 - 3 = 7 + 6 - 3 = 10 $
- $ f(n) = 2n^{2} + 7 $
$ f(1) = 2 \cdot 1^{2} + 7 = 2 + 7 = 9 $ - $ f(n) = 3n^{2} + 7 $
$ f(1) = 3 \cdot 1^{2} + 7 = 3 + 7 = 10 $ - $ f(n) = 4n^{2} + 7 $
$ f(1) = 4 \cdot 1^{2} + 7 = 4 + 7 = 11 $ - None of the above
Since b has the correct answer, it is the answer