91Ó°ÊÓ

Another form of the Binomial Theorem is \( \left(x + y\right)^n = x^n + \dfrac{nx^{n - 1}y}{1!} + \dfrac{n\left(n - 1\right)x^{n-2}y^2}{2!} + \dfrac{n\left(n - 1\right)\left(n - 2\right)x^{n-3}y^3}{3!} + \cdots + y^n \). Use this form of the Binomial Theorem to expand and simplify each expression. (a) \( \left(2 + 3\right)^6 \) (b) \( \left(x + ay\right)^4 \) (c) \( \left(x - ay\right)^5 \) (d) \( \left(1 + x\right)^{12} \)

In Exercises 93 - 106, find the sum of the infinite geometric series. \( -\dfrac{125}{36} + \dfrac{25}{6} - 5 + 6 - \cdots \)

In Exercises 107 - 110, find the rational number representation of the repeating decimal. \( 0.\overline{36} \)

In Exercises 103-112, use sigma notation to write the sum. \( 3 - 9 + 27 - 81 + 243 - 729 \)

In Exercises 103-112, use sigma notation to write the sum. \( 1 - \dfrac{1}{2} + \dfrac{1}{4} - \dfrac{1}{8} + \cdots - \dfrac{1}{128} \)

Explain how to use the first two terms of an arithmetic sequence to find the \( n \)th term.

In Exercises 107 - 110, find the rational number representation of the repeating decimal. \( 0.3\overline{18} \)

In Exercises 103-112, use sigma notation to write the sum. \( \dfrac{1}{1^2} - \dfrac{1}{2^2} + \dfrac{1}{3^2} - \dfrac{1}{4^2} + \cdots - \dfrac{1}{20^2} \)

In Exercises 107 - 110, find the rational number representation of the repeating decimal. \( 1.3\overline{8} \)

In Exercises 103-112, use sigma notation to write the sum. \( \dfrac{1}{1 \cdot 3} + \dfrac{1}{2 \cdot 4} + \dfrac{1}{3 \cdot 5} + \cdots + \dfrac{1}{10 \cdot 12} \)