Exercises

Exercises 5.5 Exercises

1.

Let \((a_n)\) be a sequence defined by \(a_n = 2^n - 1\text{.}\) Calculate \(a_5\text{.}\)

2.

Write down the first six terms of the sequence \((x_n)\) given by \(x_n=2n-3\) where \(n \ge 0\text{.}\)

3.

Write down the first five terms of the sequence \((a_n)\) where \(a_n = 2^n+n\text{,}\) \(n \ge 0\text{.}\)

4.

Let \((y_n)\) be the sequence given by the relation \(y_n = 3y_{n-1} + 2\) where \(y_0=0\text{.}\) Calculate \(y_5\) both iteratively and recursively.

5.

Consider the sequence \((a_n)\) whose terms are given by the recurrence relation

\begin{equation*} a_n = \begin{cases} 8 - 4a_{n-1} \amp n \ge 2 \\ 2 \amp n =1\end{cases} \end{equation*}

Calculate \(a_4\) both iteratively and recursively.

6.

Let \((b_n)\) be a sequence defined by

\begin{equation*} b_n = \begin{cases} 4b_{n-2} - b_{n-1} \amp n \ge 2 \\ 3 \amp n=1 \\ -1 \amp n=0 \end{cases} \end{equation*}

Compute \(b_4\) both iteratively and recursively.

7.

Consider the sequence \((x_i)\) whose terms are given by the following recurrence relation:

\begin{equation*} x_i = \begin{cases} ix_{i-1} - 2x_{i-2} \amp i \ge 2 \\ 1 \amp i = 1 \\ 1 \amp i = 0 \end{cases} \end{equation*}

Calculate \(x_4\) both iteratively and recursively.

Hint.

Hint: The coefficient \(i\) will change for each term!

8.

Let \((z_n)\) be the sequence given by the relation \(z_n=z_{n-1}z_{n-2}\) where \(z_0=2\) and \(z_1=3\text{.}\) Calculate \(z_5\) both iteratively and recursively.

9.

Consider the sequence \((y_n)\) given by the recurrence relation where \(y_1=y_2=1\) and \(y_n=ny_{n-1}y_{n-2}\) for all \(n > 2\text{.}\) Calculate \(y_5\) both iteratively and recursively.

10.

Calculate \(\displaystyle \sum_{j=1}^7 (2j-1)\text{.}\)

11.

Calculate \(\displaystyle \sum_{n=2}^5 n^3\text{.}\)

12.

Calculate \(\displaystyle \sum_{i=1}^4 i!\text{.}\)

13.

Calculate \(\displaystyle \sum_{j=2}^4 \sum_{k=5}^8 (k-j)\text{.}\)

14.

Calculate \(\sum_{m=2}^{7} \sum_{n=3}^4 (2n+m)\text{.}\)

15.

Calculate \(\displaystyle \sum_{n=0}^2 \sum_{k=1}^3 k^n\text{.}\)

16.

Calculate \(\displaystyle \sum_{m=2}^4 \sum_{n=m-1}^{m+1} mn\text{.}\)

17.

Calculate \(\sum_{m=1}^4 \sum_{n=m}^{2m} (n-2m)\text{.}\)

18.

Let \((x_n)\) be a sequence where \(\displaystyle\sum_{n=1}^{100} x_n = 50\) and \(x_{100}=12\text{.}\) Calculate

\begin{equation*} \displaystyle \sum_{n=1}^{99} 3x_{n} \end{equation*}

19.

Let \(\displaystyle \sum_{m=5}^{904} c_m = 1,307\text{,}\) \(c_4=42\text{,}\) and \(c_{904}=10\text{.}\) Compute

\begin{equation*} \displaystyle \sum_{m=4}^{903} 3c_m \end{equation*}

20.

Given that

\begin{equation*} \displaystyle \sum_{n=0}^{100} x_n = 471, \qquad \sum_{n=0}^{99} y_n = 502, \end{equation*}

and \(x_{100}=30\text{,}\) calculate

\begin{equation*} \displaystyle \sum_{n=0}^{99} (3x_n - y_n) \end{equation*}

21.

Suppose that

\begin{equation*} \sum_{n=0}^{37} a_n = 121, \qquad \sum_{n=0}^{36} b_n = 48, \end{equation*}

and furthermore \(b_{37}=10\) and \(a_{37}=-17\text{.}\) Find

\begin{equation*} \displaystyle \sum_{n=0}^{36} (2b_n+a_n) \end{equation*}