Sequences and Series

Find the first five terms of the sequence $\{a_n\}$, beginning with $n=0$:

1. $a_n = 2^n + 3$
2. $a_n = n^{n+1}$
3. $a_n = \lfloor n/2\rfloor + \lceil n/2 \rceil$
4. $a_n = 6a_{n-1},\hspace{1em} a_0 = 2$
5. $a_n = a_{n-1} + 3 a_{n-2},\hspace{1em} {a_0 = 1, a_1 = 2}$

Is the sequence $\{a_n\}$ a solution of the recurrence relation ${a_n = 8a_{n-1}-16a_{n-2}}$ if

1. $a_n = 0$
2. $a_n = 2n$
3. $a_n = 2^n$
4. $a_n = 4^n$

Find a solution to the recurrence relation

1. $a_n = 3a_{n-1},\hspace{1em} a_0 = 2$
2. $a_n = a_{n-1} + n,\hspace{1em} a_0 = 1$
3. $a_n = na_{n-1},\hspace{1em} a_0 = 1$

Find the value of each of the following sums:

1. $\displaystyle \sum_{k=0}^8 3\cdot 2^k$
2. $\displaystyle \sum_{k=1}^8 2^k$
3. $\displaystyle \sum_{k=1}^8 (1 + (-1)^k)$
4. $\displaystyle \sum_{k=1}^8 (2^{k+1} - 2^k)$
5. $\displaystyle \sum_{k=50}^{100} k$