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Chapter 14: Problem 32
Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$1^{4}+2^{4}+3^{4}+\cdots+12^{4}$$
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$$\text { Simplify: } \sqrt[3]{40 x^{4} y^{7}}$$ (Section \(10.3,\) Example 5 )
Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the exponents on \(b\).
Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 69 and 70 to verify the expansion. $$f_{1}(x)=(x+2)^{6}$$
What is the difference between a geometric sequence and an infinite geometric series?
Solve for \(P: A=\frac{P t}{P+t}\)
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