1 .
Let $n \geq 1$ be an integer, and let $S_n$ be the number of ways in which $n$ can be written
as a sum of 3's and 4's, such that the order in which the 3's and 4's occur in the sum matters.
For example, $S_5 = 0$, because 5 cannot be written as a sum of 3's and 4's.
We have $S_{11} = 3$, because
Which of the following is true for any $n \geq 5$?
| $11$ | $= 3 + 4 + 4 = 4 + 3 + 4$ $= 4 + 4 + 3.$ |
(a)
$S_n = S_{n - 3} + S_{n - 4}$
(b)
$S_n = S_{n - 1} + S_{n - 2}$
(c)
$S_n = S_{n - 2} + S_{n - 3}$
(d)
$S_n = 2 \cdot S_{n - 1}$