*label*Mathematics

### Question Description

k^{2} + 3k + 2 = (k^{2} + k) + 2 ( __________ )

## Question 2 (5 points)

The following are defined using recursion formulas. Write the first four terms of each sequence.

a_{1} = 7 and a_{n} = a_{n-1} + 5 for n ≥ 2

## Question 3 (5 points)

Write the first four terms of the following sequence whose general term is given.

a_{n} = 3n + 2

## Question 4 (5 points)

If 20 people are selected at random, ﬁnd the probability that at least 2 of them have the same birthday.

## Question 5 (5 points)

Use the Binomial Theorem to expand the following binomial and express the result in simpliﬁed form.

(x^{2} + 2y)^{4}

## Question 6 (5 points)

Consider the statement "2 is a factor of n^{2} + 3n."

If n = 1, the statement is "2 is a factor of __________."

If n = 2, the statement is "2 is a factor of __________."

If n = 3, the statement is "2 is a factor of __________."

If n = k + 1, the statement before the algebra is simpliﬁed is "2 is a factor of __________."

If n = k + 1, the statement after the algebra is simpliﬁed is "2 is a factor of __________."

## Question 7 (5 points)

Use the Binomial Theorem to expand the following binomial and express the result in simpliﬁed form.

(2x^{3} - 1)^{4}

## Question 8 (5 points)

Write the first four terms of the following sequence whose general term is given.

a_{n} = 3^{n}

## Question 9 (5 points)

Use the formula for the sum of the first n terms of a geometric sequence to solve the following.

Find the sum of the first 11 terms of the geometric sequence: 3, -6, 12, -24 . . .

## Question 10 (5 points)

A club with ten members is to choose three officers—president, vice president, and secretary-treasurer. If each office is to be held by one person and no person can hold more than one office, in how many ways can those offices be filled?

## Question 11 (5 points)

Use the Binomial Theorem to find a polynomial expansion for the following function.

f_{1}(x) = (x - 2)^{4}

## Question 12 (5 points)

Write the first four terms of the following sequence whose general term is given.

a_{n} = (-3)^{n}

## Question 13 (5 points)

You volunteer to help drive children at a charity event to the zoo, but you can ﬁt only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?

## Question 14 (5 points)

Write the first six terms of the following arithmetic sequence.

a_{1} = 5/2, d = - ½

## Question 15 (5 points)

If two people are selected at random, the probability that they do not have the same birthday (day and month) is 365/365 * 364/365. (Ignore leap years and assume 365 days in a year.)

## Question 16 (5 points)

Write the first six terms of the following arithmetic sequence.

a_{n} = a_{n-1} - 10, a_{1} = 30

## Question 17 (5 points)

Find the indicated term of the arithmetic sequence with first term, a_{1}, and common difference, d.

Find a_{200} when a_{1} = -40, d = 5

## Question 18 (5 points)

Write a formula for the general term (the n^{th} term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for a_{n} to ﬁnd a_{20}, the 20^{th} term of the sequence.

a_{n} = a_{n-1} - 10, a_{1} = 30

## Question 19 (5 points)

How large a group is needed to give a 0.5 chance of at least two people having the same birthday?

## Question 20 (5 points)

The following are defined using recursion formulas. Write the first four terms of each sequence.

a_{1} = 4 and a_{n} = 2a_{n-1} + 3 for n ≥ 2

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