Test 3 | Time: 1.5 hours |
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- Find the partial fraction decomposition of

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

**Answer:**The sequence is arithmetic, and the . - 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 - In an arithmetic sequence, and .
- Find the formula which describes the nth term of this sequence.

**Answer:**. - Using the formula found above, write the first
**three**terms of the sequence.

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

- Find the formula which describes the nth term of this sequence.
- Use mathematical induction to prove the given formula for every
positive integer n:

**Answer:**- 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*.

- When n=1, the formula is valid, because
- 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.

**Answer:**. - Describe the event that the family has exactly 2 girls.

**Answer:**.

- Describe the sample space that considers the ordering of the
births of the 3 children.
- 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).

**Answer:**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: .

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

**Answer:**In 14!=87,178,291,200 different ways. - In how many distinguishable ways can they be arranged in a row
if objects of the same shape are also the same color?

**Answer:**In different ways.

- In how many ways can she arrange them in a row if they are all
different colors?
- 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?

**Answer:**or approximately 0.3077. - 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?

**Answer:**In different ways.

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