CMSC 250 Homework 7 Fall 2001

Due Wed Oct 17 at the beginning of your discussion section.

You must write the solutions to the problems single-sided on your own lined paper, with all sheets stapled together, and with all answers written in sequential order or you will lose points.

1. Write the first 6 terms of each of the following sequences.
   1. $\forall k\geq 1,\,\,a_{k} = k^{k}$.

   2. $\forall k\geq 1,\,\,a_{k} = k^{1/k}$.

   3. $\forall k\geq 1,\,\,a_{k} = \frac{k}{k+1}$.

   4. $\forall k\geq 1,\,\,a_{k} = \frac{1}{3^{k}}$.

2. For each of the examples in number 1, write $\sum_{k = 1}^{6}a_{k}$ in expanded form and use a calculator to find the sum. You exam questions will not be this computationally intensive and will not require a calculator.

3. For each of the examples in number 1, write $\prod_{k=1}^{4}a_{k}$ in expanded form and use a calculator to find the product. Your exam questions will not be this computationally intensive and will not require a calculator.

4. For each of the following, change the following sums or products in expanded form to summation or product notation.
   1. $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}$

   2. $\frac{1}{8}\times\frac{1}{27}\times\frac{1}{64}\times\frac{1}{125}\times\frac{1}{216}\times\frac{1}{343}\times\frac{1}{512}$

   3. $\left(\frac{1}{2}\right)^{3} +\left(\frac{2}{3}\right)^{4} +\left(\frac{3}{4}\r... ...c{6}{7}\right)^{8} +\left(\frac{7}{8}\right)^{9} +\left(\frac{8}{9}\right)^{10}$

5. Use mathematical induction to prove the following formulas are valid for all integers $n$ in the specified ranges.
   1. $\forall n\geq 1,\,\,\sum_{k = 0}^{n}2\left(\frac{1}{3}\right)^{k+1} = 1-\left(\frac{1}{3}\right)^{n+1}$

   2. $\forall n\geq 1,\,\,2 + 6 + 10 + ...+ (4n-2) = 2n^{2}$.

   3. $\forall n\geq 1,\,\,\prod_{k=1}^{n}\frac{k+1}{k} = n + 1.$

   4. $\forall n\geq 6,\,\,n^{2} > 3n+12$.

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