91Ó°ÊÓ

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{\mathrm{MMM},\) MMF. MFM. MFF. FMM. FMF, FFM, FFF . Find the probability of selecting a family with at least one male child.

Evaluate each factorial expression. $$\frac{17 !}{15 !}$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(c+3)^{5}$$

In Exercises \(23-34,\) write a formula for the general term (the nth 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. $$2,7,12,17, \dots$$

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{\mathrm{MMM},\) MMF. MFM. MFF. FMM. FMF, FFM, FFF . Find the probability of selecting a family with at least two female children.

Evaluate each expression. $$ 1-\frac{_5 P_{3}}{_{10} P_{4}} $$

Evaluate each factorial expression. $$\frac{18 !}{16 !}$$

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$0.0007,-0.007,0.07,-0.7, \ldots$$

In Exercises 25-34, use mathematical induction to prove that each statement is true for every positive integer \(n.\) 2 is a factor of \(n^{2}-n\).