91Ó°ÊÓ

Show that a right triangle whose sides are in arithmetic progression is similar to a \(3-4-5\) triangle.

Find the sum of the infinite geometric series. $$3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+\dots$$

Add each of the first five rows of Pascal's triangle, as indicated. Do you see a pattern? $$\begin{aligned} &\begin{array}{c} 1+1=? \\ 1+2+1=? \end{array}\\\ &1+3+3+1=?\\\ &1+4+6+4+1=?\\\ &1+5+10+10+5+1=? \end{aligned}$$Based on the pattern you have found, find the sum of the \(n\) th row: $$ \left(\begin{array}{l} n \\ 0 \end{array}\right)+\left(\begin{array}{l} n \\ 1 \end{array}\right)+\left(\begin{array}{l} n \\ 2 \end{array}\right)+\cdots+\left(\begin{array}{l} n \\ n \end{array}\right) $$ Prove your result by expanding \((1+1)^{n}\) using the Binomial Theorem.

Find the product of the numbers \(10^{1 / 10}, 10^{2 / 10}, 10^{3 / 10}, 10^{4 / 10}, \ldots, 10^{19 / 10}\)

Use a graphing calculator to evaluate the sum. $$\sum_{n=1}^{100} \frac{(-1)^{n}}{n}$$

A sequence is harmonic if the reciprocals of the terms of the sequence form an arithmetic sequence. Determine whether the following sequence is harmonic: $$1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \dots$$

Find the sum of the infinite geometric series. $$-\frac{100}{9}+\frac{10}{3}-1+\frac{3}{10}-\dots$$

Write the sum without using sigma notation. $$\sum_{k=1}^{5} \sqrt{k}$$

The harmonic mean of two numbers is the reciprocal of the average of the reciprocals of the two numbers. Find the harmonic mean of 3 and 5

Write the sum without using sigma notation. $$\sum_{i=0}^{4} \frac{2 i-1}{2 i+1}$$