We need to find the first eight terms of the Fibonacci sequence where $u_1 = 1$ and $u_2 = 1$, with the recurrence relation $u_n = u_{n-1} + u_{n-2}$. We also need to find the first five terms of the sequence defined by $u_n = \frac{(-1)^{n+1}}{n}$.

1. Problem Description

We need to find the first eight terms of the Fibonacci sequence where $u_1 = 1$ and $u_2 = 1$, with the recurrence relation $u_n = u_{n-1} + u_{n-2}$.

We also need to find the first five terms of the sequence defined by $u_n = \frac{(-1)^{n+1}}{n}$.

2. Solution Steps

(c) Fibonacci Sequence:

We are given $u_1 = 1$ and $u_2 = 1$. We use the recurrence relation $u_n = u_{n-1} + u_{n-2}$ to find the next terms.

$u_3 = u_2 + u_1 = 1 + 1 = 2$

$u_4 = u_3 + u_2 = 2 + 1 = 3$

$u_5 = u_4 + u_3 = 3 + 2 = 5$

$u_6 = u_5 + u_4 = 5 + 3 = 8$

$u_7 = u_6 + u_5 = 8 + 5 = 13$

$u_8 = u_7 + u_6 = 13 + 8 = 21$

Therefore, the first eight terms are 1, 1, 2, 3, 5, 8, 13,

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1.

(d) Sequence $u_n = \frac{(-1)^{n+1}}{n}$:

We plug in $n = 1, 2, 3, 4, 5$ to find the first five terms.

$u_1 = \frac{(-1)^{1+1}}{1} = \frac{(-1)^2}{1} = \frac{1}{1} = 1$

$u_2 = \frac{(-1)^{2+1}}{2} = \frac{(-1)^3}{2} = \frac{-1}{2} = -\frac{1}{2}$

$u_3 = \frac{(-1)^{3+1}}{3} = \frac{(-1)^4}{3} = \frac{1}{3}$

$u_4 = \frac{(-1)^{4+1}}{4} = \frac{(-1)^5}{4} = \frac{-1}{4} = -\frac{1}{4}$

$u_5 = \frac{(-1)^{5+1}}{5} = \frac{(-1)^6}{5} = \frac{1}{5}$

Therefore, the first five terms are $1, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{4}, \frac{1}{5}$.

3. Final Answer

(c) The first eight terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13,

2

1. (d) The first five terms of the sequence $u_n = \frac{(-1)^{n+1}}{n}$ are $1, -\frac{1}{2}, \frac{1}{3}, -\frac{1}{4}, \frac{1}{5}$.