91Ó°ÊÓ

The following formulas can be used to find sums of powers of natural numbers. Use mathematical induction to prove each formula. $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$

Find the first term and the common difference. Find \(a_{1}\) when \(d=4\) and \(a_{8}=33\)

Expand. \(\left(x^{-2}+x^{2}\right)^{4}\)

The following formulas can be used to find sums of powers of natural numbers. Use mathematical induction to prove each formula. $$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4}$$

Use a graphing calculator to construct a table of values and a graph for the first 10 terms of the sequence. $$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$

An American roulette wheel contains 38 slots numbered \(00,0,1,2,3, \ldots, 35,36 .\) Eighteen of the slots numbered \(1-36\) are colored red and 18 are colored black. The 00 and 0 slots are considered to be uncolored. The wheel is spun, and a ball is rolled around the rim until it falls into a slot. What is the probability that the ball falls in each of the following? A red slot

The following formulas can be used to find sums of powers of natural numbers. Use mathematical induction to prove each formula. $$\begin{aligned} 1^{4}+& 2^{4}+3^{4}+\cdots+n^{4} \\ &=\frac{n(n+1)(2 n+1)\left(3 n^{2}+3 n-1\right)}{30}\end{aligned}$$

Find the first term and the common difference. Find \(d\) when \(a_{1}=8\) and \(a_{11}=26\)

Use a graphing calculator to construct a table of values and a graph for the first 10 terms of the sequence. $$a_{n}=\sqrt{n+1}-\sqrt{n}$$

Expand. \(\left(\frac{1}{\sqrt{x}}-\sqrt{x}\right)^{6}\)