Prove Euler's Formula for graph inductively.

1. How many 12-bit strings (that is, bit strings of length 14) start with the sub-string 011?

2. You break your piggy bank to discover lots of pennies and nickels.

You start arranging them in rows of 6 coins.

How many coins would you need to make all possible rows of 6 coins (not necessarily with equal numbers

of pennies and nickels)?

3. How many shortest lattice paths start at (4, 4) and end at (13, 13)?

4. What is the coefficient of x⁵y² in the expansion of (x + y)⁷?

1. Express the function f(t) = 0, -π/2<t<π/2

                                           = 1, -π<t<-π/2 and π/2<t<π

with f(t+2π)=f(t), as a Fourier series.

Let sn = 2^1/n+ n sin(nπ/2), n ∈ N.

(a) List all subsequential limits of (sn).

(b) Give a formula for nk such that (snk) is an unbounded increasing subsequence of (sn).

(c) Give a formula for nk such that (snk) is a convergent subsequence of (sn).

For each sequence given below, find a closed formula for an, the nth term of the sequence (assume the first terms here are always a0) by relating it to another sequence for which you already know the formula.

−1,0,7,26,63,124,…

an=(n^3)-1

−1,1,7,17,31,49,…

an=2n^2-1

0,10,30,60,100,150,.....

an=10*((n(n+1))/2)

2,3,6,14,40,152,…

an=

The first three are correct I can't figure out the last one.

solve the equation using the variable separation method

u_tt = u_xx + a
0< x< 1, t> 0
with boundary contions:
u(0,t)=u(1,0)=0
u(x,0)=mx(1-x)
u_t(x,0)=0

Dr. Lori Baker, operations manager at Nesa Electronics, prides herself on excellent assembly-line balancing. She has been told that the firm needs to complete 96 instruments per 24-hour day. The assembly-line activities are: TASK. TIME (min). PREDECESSORS. A 3 — B 6 — C 7 A D 5 A, B E 2 B F 4 C G 5 F H 7 D, E I 1 H J 6 E K 4 G, I, J Total 50 Draw the precedence diagram. If the daily (24-hour) production rate is 96 units, what is the highest allowable cycle time? minutes. If the cycle time after allowances is given as 10 minutes, what is the daily (24-hour) production rate? units per day. With a 10-minute cycle time, what is the theoretical minimum number of stations with which the line can be balanced? With a 10-minute cycle time and six workstations, what is the efficiency? % What is the total idle time per cycle with a 10-minute cycle time and six workstations? minutes.

Prove that the rational numbers do not satisfy the least upper bound axiom. In particular, if a subset (S) of the rational numbers is bounded above and M is the set of all rational upper bounds of S, then M may not have a least element.

A foundry that specializes in producing custom blended alloys has received an order for 1 000 kg of an alloy containing at least 5% chromium and not more than 50% iron. Four types of scrap which can be easily acquired can be blended to produce the order. The cost and metal characteristics of the four scrap types are given below: Scrap type Item 1 2 3 4 Chromium 5% 4% - 8% Iron 40% 80% 60% 32% Cost per kg R6 R5 R4 R7 The purchasing manager has formulated the following LP model: Minimise COST = 6M1 + 5M2 + 4M3 + 7M4 subject to 0,05M1 + 0,04M2 + 0,08M4 ≥ 50 (CHRM) 0,40M1 + 0,80M2 + 0,60M3 + 0,32M4 ≤ 500 (IRON) M1 + M2 + M3 + M4 = 1000 (MASS) and all variables ≥ 0, where Mi = number of kg of scrap type i purchased, i=1,2,3,4. (a) Solve this model using LINGO or SOLVER. (b) Write down the foundry's optimal purchasing plan and cost. Based on your LINDO or SOLVER solution answer the following questions by using only the initial printout of the optimal solution. (This means that you may not change the relevant parameters in the model and do reruns.) (c) How good a deal would the purchasing manager need to get on scrap type 1 before he would be willing to buy it for this order? (d) Upon further investigation, the purchasing manager finds that scrap type 2 is now being sold at R5,40 per kg. Will the purchasing plan change? By how much will the cost of purchasing the metals increase? (e) The customer is willing to raise the ceiling on the iron content in order to negotiate a reduction in the price he pays for the order. How should the purchasing manager react to this? (f) The customer now specifies that the alloy must contain at least 6% chromium. Can the purchasing manager comply with this new specification? Will the price charged for the order change?

For the following ODE

4. xy'' +y' -xy = 0; x₀ = 0

a) What are the points of singularity for the problem?

b) Does this ODE hold a general power series solution at the specific x0? Justify your answer,

i) if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and Identify the recursion formula for the power series coefficient around x0, and write the corresponding solution with at least four non-zero terms explicitly written

ii) If your answer is no, then please elaborate why there is not a general power series solution, and proceed to solve it using the Frobenius method

Prove:

If a1 = b1 mod n and a2 = b2 mod n then

(1) a1 + a2 = b1 + b2 mod n,

(2) a1 − a2 = b1 − b2 mod n, and

(3) a1a2 = b1b2 mod n.

5. Individual Problems 16-5

Your pharmaceutical firm is seeking to open up new international markets by partnering with various local distributors. The different distributors within a country are stronger with different market segments (hospitals, retail pharmacies, etc.) but also have substantial overlap.

In Egypt, you calculate that the annual value created by one distributor is $90 million per year, but would be $120 million if two distributors carried your product line.

Assuming a nonstrategic view of bargaining, you would expect to capture $______ million of this deal. (Hint: The two distributors are independent of each other; therefore, you conduct separate negotiations with each.)

Argentina also has two distributors that add value equivalent to the value added by the two distributors in Egypt, but both are run by the government.

Assuming a nonstrategic view of bargaining, you would expect to capture $______ million of this deal.

In Argentina, if you do not reach an agreement with the government distributors, you can set up a less efficient Internet-based distribution system that would generate $30 million in value to you.

Assuming a nonstrategic view of bargaining, you would expect to capture $______ million of this deal.

Which positive integers n, where 20 ≤ n ≤ 30, have primitive roots?

Find a primitive root modulo 2401 = 7^4. Be sure to mention which exponentiations you checked to prove that your final answer is indeed a primitive root. (You may use Wolfram Alpha for exponentiations modulo 2401, but you may not use any of Wolfram Alpha’s more powerful functions.)