91Ó°ÊÓ

Solve the recurrence relation \(a_{n}=3 a_{n-1}+10 a_{n-2}\) with initial terms \(a_{0}=4\) and \(a_{1}=1\)

The in song The Twelve Days of Christmas, my true love gave to me first 1 gift, then 2 gifts and 1 gift, then 3 gifts, 2 gifts and 1 gift, and so on. How many gifts did my true love give me all together during the twelve days?

Prove that the sum of \(n\) squares can be found as follows $$ 1^{2}+2^{2}+3^{2}+\ldots+n^{2}=\frac{n(n+1)(2 n+1)}{6} $$

Consider the sequence \(2,7,15,26,40,57, \ldots\) (with \(a_{0}=2\) ). By looking at the differences between terms, express the sequence as a sequence of partial sums. Then find a closed formula for the sequence by computing the \(n\) th partial sum.

Prove that the sum of the interior angles of a convex \(n\) -gon is \((n-2) \cdot 180^{\circ} .\) (A convex \(n\) -gon is a polygon with \(n\) sides for which each interior angle is less than \(180^{\circ} .\) )

Will the \(n\) th sequence of differences of \(2,6,18,54,162, \ldots\) ever be constant? Explain.

Starting with any rectangle, we can create a new, larger rectangle by attaching a square to the longer side. For example, if we start with a \(2 \times 5\) rectangle, we would glue on a \(5 \times 5\) square, forming a \(5 \times 7\) rectangle: The next rectangle would be formed by attaching a \(7 \times 7\) square to the top or bottom of the \(5 \times 7\) rectangle. (a) Create a sequence of rectangles using this rule starting with a \(1 \times 2\) rectangle. Then write out the sequence of perimeters for the rectangles (the first term of the sequence would be \(6,\) since the perimeter of a \(1 \times 2\) rectangle is 6 - the next term would be 10 ). (b) Repeat the above part this time starting with a \(1 \times 3\) rectangle. (c) Find recursive formulas for each of the sequences of perimeters you found in parts (a) and (b). Don't forget to give the initial conditions as well. (d) Are the sequences arithmetic? Geometric? If not, are they close to being either of these (i.e., are the differences or ratios almost constant)? Explain.

Solve the recurrence relation \(a_{n}=2 a_{n-1}-a_{n-2}\) (a) What is the solution if the initial terms are \(a_{0}=1\) and \(a_{1}=2 ?\) (b) What do the initial terms need to be in order for \(a_{9}=30 ?\) (c) For which \(x\) are there initial terms which make \(a_{9}=x ?\)

Use summation \(\left(\sum\right)\) or product \(\left(\prod\right)\) notation to rewrite the following. (a) \(2+4+6+8+\cdots+2 n\). (d) \(2 \cdot 4 \cdot 6 \cdots \cdot \cdot 2 n\) (b) \(1+5+9+13+\cdots+425\) (c) \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{50}\). (e) \(\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\left(\frac{3}{4}\right) \cdots\left(\frac{100}{101}\right)\)

If you have enough toothpicks, you can make a large triangular grid. Below, are the triangular grids of size 1 and of size 2 . The size 1 grid requires 3 toothpicks, the size 2 grid requires 9 toothpicks. (a) Let \(t_{n}\) be the number of toothpicks required to make a size \(n\) triangular grid. Write out the first 5 terms of the sequence \(t_{1}, t_{2}, \ldots\) (b) Find a recursive definition for the sequence. Explain why you are correct. (c) Is the sequence arithmetic or geometric? If not, is it the sequence of partial sums of an arithmetic or geometric sequence? Explain why your answer is correct. (d) Use your results from part (c) to find a closed formula for the sequence. Show your work.