(1) The first question defines a sequence where each term represents the vertical bounce of a ball. It asks to write the first 5 terms and the total distance bounced when the ball comes to rest. (2) The second question asks to use mathematical induction to prove an equation involving a summation and fractions. (3) The third question uses the binomial expansion theorem to expand a term and asks for the first four terms as percentages and which represents the probability of all tails. (4) The fourth question expands another term using the binomial expansion theorem. (5) The fifth question asks to prove an identity involving binomial coefficients using induction or an formula.

1. Math 4R Test12A Name:
For each question: Show all work. Justify each answer in the space provided. Circle final answers.
  3 n-1
(1) Let the sequence terms of {an} = 20   represent the
  4 
vertical distance a ball bounces measured in feet. For example,
a1 = 20’ means that the ball bounces up 10’ and then falls 10’ before the
second bounce a2.
(1a) Write the first 5 terms of the sequence {an}.
(1b) What is the total vertical distance the ball bounced once it has come to
rest?
a:TEST12A.4Rt

2. Math 4R Test12A Name:
For each question: Show all work. Justify each answer in the space provided. Circle final answers.
n n2(n+1)2
(2) Use Mathematical Induction to prove: i3 = 4
i=1
a:TEST12A.4Rt

3. Math 4R Test12A Name:
For each question: Show all work. Justify each answer in the space provided. Circle final answers.
(3) Use the Binomial Expansion Theorem to expand:
3
(0.5 + 0.5)3 =  nCr (0.5)3-r (0.5)r
r=0
(3a) Write the first term of this expansion as a percent.
(3b) Write the second term of this expansion as a percent.
(3c) Write the third term of this expansion as a percent.
(3d) Write the fourth term of this expansion as a percent.
(3e) Which of these percents represents the probability of getting only tails?
Hint: (3C1) (0.5)2 (0.5)1 represents the probability of getting 2 heads
when flipping a fair coin 3 times.
a:TEST12A.4Rt

4. Math 4R Test12A Name:
For each question: Show all work. Justify each answer in the space provided. Circle final answers.
(4) Use the Binomial Expansion Theorem to expand:
(5 – 3i)4
a:TEST12A.4Rt

5. Math 4R Test12A Name:
For each question: Show all work. Justify each answer in the space provided. Circle final answers.
(5) Prove: nCm = nCn-m
Hint: Use Mathematical Induction or the formula for nCr.
a:TEST12A.4Rt