Difference between revisions of "2013 AMC 12A Problems/Problem 7"

Problem

The sequence $S_1, S_2, S_3, \cdots, S_{10}$ has the property that every term beginning with the third is the sum of the previous two. That is, $$S_n = S_{n-2} + S_{n-1} \text{ for } n \ge 3.$$ Suppose that $S_9 = 110$ and $S_7 = 42$. What is $S_4$?

$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 16\qquad$

Solution

$S_9 = 110$, $S_7 = 42$

$S_8 = S_9 - S_ 7 = 110 - 42 = 68$

$S_6 = S_8 - S_7 = 68 - 42 = 26$

$S_5 = S_7 - S_6 = 42 - 26 = 16$

$S_4 = S_6 - S_5 = 26 - 16 = 10$

Therefore, the answer is $\boxed{\textbf{(C) }{10}}$

Video Solution

https://youtu.be/CCjcMVtkVaQ ~sugar_rush