91Ó°ÊÓ

Use mathematical induction to prove that each statement is true for every positive integer. $$\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\frac{1}{4 \cdot 5}+\dots+\frac{1}{(n+1)(n+2)}=\frac{n}{2 n+4}$$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(c+3)^{5}$$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-1)^{5}$$

Evaluate each factorial expression. $$\frac{16 !}{2 ! 14 !}$$

Evaluate each expression. $$\frac{_{7} C_{3}}{_{5} C_{4}}-\frac{98 !}{96 !}$$

Use mathematical induction to prove that each statement is true for every positive integer \(n\). 2 is a factor of \(n^{2}-n\)

Use the formula for the sum of the first \(n\) terms of a geometric sequence to solve. Find the sum of the first 12 terms of the geometric sequence: \(2,6,18,54, \dots\)

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. $$7,3,-1,-5, \dots$$

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. $$6,1,-4,-9, \dots$$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x-2)^{5}$$