A6 / B7

Part of 2011 Princeton University Math Competition

Problems(2)

2011 PUMaC Algebra A6 / B7

Source:

A sequence of real numbers

\{a_n\}_{n = 1}^\infty (n=1,2,...)

has the following property: \begin{align*} 6a_n+5a_{n-2}=20+11a_{n-1}\ (\text{for }n\geq3). \end{align*} The first two elements are

a_1=0, a_2=1

. Find the integer closest to

a_{2011}

2011 PUMaC Combinatorics A6 / B7

Source:

For every integer

n

0

6

3

identical weights with weight

2^n

. How many ways are there to form a total weight of 263 grams using only these given weights?