Transform each by making the change of variable i=k+1.

$$\prod _ { k = 1 } ^ { n } \frac { k } { k ^ { 2 } + 4 }$$

Explain the importance of each of the following terms. Red Army, supra.

In 1927 , the ophthalmologist George Waaler tested 9049 schoolboys in Oslo, Norway, for red-green color blindness and found 8324 of them to be normal and 725 to be color blind. He also tested 9072 schoolgirls and found 9032 that had normal color vision while 40 were color blind.

a. Assuming that the same sex-linked recessive allele $c$ causes all forms of red-green color blindness, calculate the allele frequencies of $c$ and $C$ (the allele for normal vision) from the data for the schoolboys. (Hint: Refer to your answer to Problem 12a.)

b. Does Waaler's sample demonstrate Hardy-Weinberg equilibrium for alleles of this gene? Explain your answer by describing observations that are either consistent or inconsistent with this hypothesis. On closer analysis of these schoolchildren, Waaler found that there was actually more than one $c$ allele causing color blindness in his sample: one kind for the prot type $\left(c^p\right)$ and one for the deuter type $\left(c^d\right)$. (Protanopia and deuteranopia are slightly different forms of red-green color blindness.) Importantly, some of the apparently normal females in Waaler's studies were probably of genotype $c^p / c^d$. Through further analysis of the 40 color-blind females, he found that 3 were prot $\left(c^p / c^p\right)$, and 37 were deuter $\left(c^d / c^d\right)$.

c. Based on this new information, what are the frequencies of the $c^p, c^d$, and $C$ alleles in the population examined by Waaler? Calculate these values as if the frequencies obey the Hardy-Weinberg equilibrium. (Note: Again, refer to your answer to Problem 12a.)

d. Calculate the frequencies of all genotypes expected among men and women if the population is at equilibrium.

e. Do these results make it more likely or less likely that the population in Oslo is indeed at equilibrium for red-green color blindness? Explain your reasoning.

Evaluate the summations and products for the indicated values of the variable.

$$1 ( 1 ! ) + 2 ( 2 ! ) + 3 ( 3 ! ) + \ldots + m ( m ! ) ; m = 2$$

Evaluate the following without using a calculator. $\operatorname{arcsec}(\sec \pi)$

Prove that if m and n are integers and mn is even, then m is even or n is even.

Prove that for all integers m and n, if m and m+n are even, then n is even.

Prove each of the statements in two ways: (a) by contraposition and (b) by contradiction. For all integers m and n, if mn is even then m is even or n is even.

Expectant parents are thrilled to hear their unborn baby’s heartbeat, revealed by an ultrasonic motion detector. Suppose the fetus’s ventricular wall moves in simple harmonic motion with amplitude 1.80 mm and frequency 115 beats per minute. The motion detector in contact with the mother’s abdomen produces sound at precisely 2 MHz, which travels through tissue at 1.50 km/s. Find the maximum linear speed of the heart wall.

An ice cube of mass 9.0 g is added to a cup of coffee. The coffee's initial temperature is 90.0$^{\circ} C$ and the cup contains 120.0 g of liquid. Assume the specific heat capacity of the coffee is the same as that of water. The heat of fusion of ice (the heat associated with ice melting) is 6.0 kJ/mol. Find the temperature of the coffee after the ice melts.

A newly issued bond pays its coupons once a year. Its coupon rate is 5%, its maturity is 20 years, and its yield to maturity is 8%. What is the after-tax holding-period return on the bond?

Is the limit of f (x) as x approaches c always equal to f (c)? Why or why not?

Calculate:

$$\frac{13}{18}\cdot\frac{25}{7}\cdot\frac{18}{13}$$

By what process is a prisoner granted parole?