Exercises 6.5 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
Calculate \(a_4\) both iteratively and recursively.
6.
Let \((b_n)\) be a sequence defined by
Compute \(b_4\) both iteratively and recursively.
7.
Consider the sequence \((x_i)\) whose terms are given by the following recurrence relation:
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
19.
Let \(\displaystyle \sum_{m=5}^{904} c_m = 1,307\text{,}\) \(c_4=42\text{,}\) and \(c_{904}=10\text{.}\) Compute
20.
Given that
and \(x_{100}=30\text{,}\) calculate
21.
Suppose that
and furthermore \(b_{37}=10\) and \(a_{37}=-17\text{.}\) Find