1
a) Observe that 1+2 = 3 = 2^2-1, 1+2+2^2= 7 = 2^3-1, 1+2+2^2+2^3 =
15...
1
a) Observe that 1+2 = 3 = 2^2-1, 1+2+2^2= 7 = 2^3-1, 1+2+2^2+2^3 =
15 = 2^4−1, so it appears to be the case that 1+2+2^2+2^3+···+2n =
2n+1 − 1 for any integer n ≥ 0. Use induction on n to prove that
this is true.
b) Find all three solutions of the equation 2z^3 + 4z^2 − z −
5 = 0.
(First try a few “simple” values of z)
1) determine the 30th term
15, -85, -185, 285...
2) determine the 10th term
-4, 8,...
1) determine the 30th term
15, -85, -185, 285...
2) determine the 10th term
-4, 8, -16, 32...
3) use the equation to complete the statement
a(n)=14+2(n-1)
the commone differerence is ____ and the first term is
____
4) use the equation to complete the statement
a(1)= -3; a(n)= -2a(n-1)
the common ratio is ____ and the first term is ____
5) a(n)= -5( n-1/n+2) +4
thw given equation is (arithmetic, geometric, neither)
for the following, write an explicit formula...
Consider the following:
period 1, 2, 3, 4, 5, 6, 7, 8
demand 7, 8, 9,...
Consider the following:
period 1, 2, 3, 4, 5, 6, 7, 8
demand 7, 8, 9, 10, 14, 16, 13, 16
a. using a trend projection, forecast the demand for period
9
b. calculate the MAD for this forecast
Show all work! do not use excel or phstat!!!
Consider the following recursive equation s(2n) = 2s(n) + 3;
where n = 1, 2, 4,...
Consider the following recursive equation s(2n) = 2s(n) + 3;
where n = 1, 2, 4, 8, 16, ...
s(1) = 1
a. Calculate recursively s(8)
b. Find an explicit formula for s(n)
c. Use the formula of part b to calculate s(1), s(2), s(4), and
s(8)
d Use the formula of part b to prove the recurrence equation
s(2n) = 2s(n) + 3
Simplify the following Boolean functions, using K-maps. Find all
the prime implicants, and determine which are...
Simplify the following Boolean functions, using K-maps. Find all
the prime implicants, and determine which are essential:
(a) F (w, x, y, z) = ? (1, 4, 5, 6, 12, 14, 15)
(b) F (A, B, C, D) = ? (2, 3, 6, 7, 12, 13, 14)
(c) F (w, x, y, z) = ? (1, 3, 4, 5, 6, 7, 9, 11, 13, 15)
3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine...
3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.