91Ó°ÊÓ

Critical Thinking Find the specified value for each infinite geometric series. $$ a_{1}=12, S=96 ; \text { find } r $$

a. Use your calculator to generate an arithmetic sequence with a common difference of - \(7 .\) How could you se a calculator to find the 6th term? The 8th term? The 20th term? b. Explain how your answer to part (a) relates to the explicit formula \(a_{n}=a_{1}+(n-1) d\)

a. Open-Ended Write four terms of a sequence of numbers that you can describe both recursively and explicitly. b. Write a recursive formula and an explicit formula for your sequence. c. Find the 20 th term of the sequence by evaluating one of your formulas. Use the other formula to check your work.

The sum of an infinite geometric series is twice its first term. a. Error Analysis A student says the common ratio of the series is \(\frac{3}{2} \cdot\) What is the student's error? b. Find the common ratio of the series.

Find the 17th term of each sequence. \(a_{16}=18, d=\frac{1}{2}\)

The area under a curve is estimated using inscribed rectangles and circumscribed rectangles. Explain why the mean of these two values might be a more accurate estimate than either one.

What is the common ratio for the geometric series \(\sum_{n=1}^{10} 7\left(\frac{4}{7}\right)^{n-1} ?\) Enter your answer as a fraction.

Write the equation of each hyperbola in standard form. Sketch the graph. $$ 9 x^{2}-16 y^{2}=144 $$

Geometry The triangular numbers form a sequence. The diagram represents the first three triangular numbers: \(1,3,\) and \(6 .\) a. Find the fifth and sixth triangular numbers. b. Write a recursive formula for the \(n\) th triangular number. c. Is the explicit formula \(a_{n}=\frac{1}{2}\left(n^{2}+n\right)\) the correct formula for this sequence? How do you know?

What is the product of the geometric mean of 2 and 32 and the geometric mean of 1 and 4\(?\) $$ \begin{array}{lllll}{\text { F. } 16} & {\text { G. } 19} & {\text { H. } 32} & {\text { 1. } 256}\end{array} $$