## Precalculus2 Practice Exam

Test 3 Time: 1.5 hours

1. Find the partial fraction decomposition of

2. Identify the sequence (in the sum) as arithmetic, geometric, or neither and find the sum:

Answer: The sequence is arithmetic, and the .

3. Identify the sequence as arithmetic, geometric, or neither and find the sum of the first 25 terms. Round your answer to two decimal places:

Answer: The sequence is geometric, and the sum equals

4. In an arithmetic sequence, and .
• Find the formula which describes the nth term of this sequence.

• Using the formula found above, write the first three terms of the sequence.

Answer: The first three terms are 8, 5, 2.

5. Use mathematical induction to prove the given formula for every positive integer n:

• When n=1, the formula is valid, because

• Assuming that

show that

To do this, write the following:

Combining the results of the two parts, it can be concluded by mathematical induction that the formula is valid for every positive integer n.

6. An experiment consists of studying the number of boys and girls in families with exactly 3 children. Let b represent boys and g represent girls.
• Describe the sample space that considers the ordering of the births of the 3 children.

• Describe the event that the family has exactly 2 girls.

7. Consider the expression .
• Using the Binomial Theorem, write out the binomial expansion of the given expression and compute the third binomial coefficient by hand (show all steps).

the third binomial coefficient is equal to 6.

• Using any method (calculator or by hand) find the rest of the coefficients, and simplify the expansion in the above part.

Answer: Coefficients are: 1, 4, 6, 4, and 1, and simplification of the expression yields: .

8. A child has a set of different shaped plastic objects. There are 3 pyramids, 4 cubes, and 7 spheres.
• In how many ways can she arrange them in a row if they are all different colors?

• In how many distinguishable ways can they be arranged in a row if objects of the same shape are also the same color?

9. One card is selected from a standard deck of 52 playing cards. What is the probability that the card is a heart or a two?

10. From a group of 16 smokers and 20 nonsmokers, a researcher wants to randomly select 8 smokers and 8 nonsmokers for a study. In how many ways can the study group be selected?