A person deposits $1000 in an account that yields 9% interest compounded annually. a) Set up a recurrence relation for the amount in the account at the end of n years. b) Find an explicit formula for the amount in the account at the end of n years. c) How much money will the account contain after 100 years?

What are the values of these sums, where $S$={1,3,5,7}? a) $\sum_{j\in S} j$ b) $\sum_{j\in S} j^2$ c) $\sum_{j\in S} (1/j)$ d) $\sum_{j\in S}1$

Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. a) $a_n = −2a_{n−1}, a_0 = −1$. b) $a_n = a_{n−1} − a_{n−2}, a_0 = 2, a_1 = −1$. c) $a_n = 3a^2_{n−1}, a_0 = 1$. d) $a_n = na_{n−1} + a^2_{n−2}, a_0 = −1, a_1 = 0$. e) $a_n = a_{n−1} − a_{n−2} + a_{n−3}, a_0 = 1, a_1 = 1, a_2 = 2$

Suppose that f(n) = f(n/5) + 3n² when n is a positive integer divisible by 5, and f(1) = 4. Find a) f(5). b) f(125). c) f(3125).

Draw the graph of the function f(x) = ⌊2x⌋ from R to R.

Find $\prod_{j=0}^4 j!$.

List the first 10 terms of each of these sequences. a) the sequence that begins with 2 and in which each successive term is 3 more than the preceding term. b) the sequence that lists each positive integer three times, in increasing order. c) the sequence that lists the odd positive integers in increasing order, listing each odd integer twice. d) the sequence whose nth term is n! − 2ⁿ. e) the sequence that begins with 3, where each succeeding term is twice the preceding term. f) the sequence whose first term is 2, second term is 4, and each succeeding term is the sum of the two preceding terms. g) the sequence whose nth term is the number of bits in the binary expansion of the number n. h) the sequence where the nth term is the number of letters in the English word for the index n.

For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. a) 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, . . . b) 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, . . . c) 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, . . . d) 1, 2, 2, 2, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, . . . e) 0, 2, 8, 26, 80,242,728,2186,6560,19682, . . . f) 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, . . . g) 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, . . . h) 2, 4, 16, 256, 65536, 4294967296, . . .

Find the value of each of these sums:

a) $\sum_{j=0}^8 (1+(-1))^i$

b) $\sum_{j=0}^8 (3^j-2^j)$

c) $\sum_{j=0}^8 (2\cdot 3^j+3\cdot 2^j)$

d) $\sum_{j=0}^8(2^{j+1}-2^j)$

Find the first five terms of the sequence defined by each of these recurrence relations and initial conditions. a) $a_n = 6a_{n−1}, a_0= 2$. b) $a_n = a^2_{n−1}, a_1 = 2$. c) $a_n = a_{n−1} + 3a_{n−2}, a_0 = 1, a_1 = 2$. d) $a_n = na_{n−1} + n^2a_{n−2}, a_0 = 1, a_1 = 1$. e) $a_n = a_{n−1} + a_{n−3}, a_0 = 1, a_1 = 2, a_2 = 0$

Find the successors of the following sets. a) {1, 2, 3} b) ∅ c) {∅} d) {∅, {∅}}

Determine whether f : Z × Z → Z is onto if a) f (m, n) = 2m − n. b) f (m, n) = m² − n². c) f (m, n) = m + n + 1. d) f (m, n) = |m| − |n|. e) f (m, n) = m² − 4.

What are the values of these sums? a) $\sum_{k=1}^n (k+1)$ b) $\sum_{j=0}^4 (-2)^j$ c) $\sum_{i=1}^{10} 3$ d) $\sum_{j=0}^8 (2^{j+1}-2^j)$

Solve these recurrence relations together with the initial conditions given. a) $a_n = a_{n−1} + 6a_{n−2}$ for $n \geq 2$, $a_0 = 3, a_1 = 6$. b) $a_n = 7a_{n−1} − 10a_{n−2}$ for $n \geq 2$, $a_0 = 2, a_1 = 1.$c) $a_n = 6a_{n−1} − 8a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 10$. d) $a_n = 2a_{n−1} − a_{n−2}$ for $n \geq 2, a_0 = 4, a_1 = 1$. e) $a_n = a_{n−2}$ for $n \geq 2, a_0 = 5, a_1 = −1$. f) $a_n = −6a_{n−1} − 9a_{n−2}$ for $n \geq 2, a_0 = 3, a_1 = −3$. g) $a_{n+2} = −4a_{n+1} + 5a_n$ for $n \geq 0, a_0 = 2, a_1 = 8$