Question

identify the explicit formula for the sequence given by the following recursive formula: $f(n)=\begin{cases}f(1)= - 2\\f(n)=f(n - 1)+4\text{ if }n>1end{cases}$
a) $f(n)=2 - 4(n - 1)$
b) $f(n)=-2 + 4(n - 1)$
c) $f(n)=-4 + 2(n - 1)$
d) $f(n)=4 - 2(n - 1)$
question 4 (5 points)
find the sum of the finite sequence: 4, 2, 0, -2, - 4.
a) 1
b) 4

Explanation:

Step1: Analyze the recursive formula

The recursive formula $f(1)= - 2$ and $f(n)=f(n - 1)+4$ for $n>1$ represents an arithmetic - sequence with first term $a_1=-2$ and common difference $d = 4$. The explicit formula for an arithmetic sequence is $f(n)=a_1+(n - 1)d$.

Step2: Substitute values into the formula

Substitute $a_1=-2$ and $d = 4$ into the explicit - formula $f(n)=a_1+(n - 1)d$. We get $f(n)=-2+4(n - 1)$.

Step3: Analyze the sum of the sequence $4,2,0,-2,-4$

The sum of a finite arithmetic sequence $S_n=\frac{n(a_1 + a_n)}{2}$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. Here, $n = 5$, $a_1=4$, and $a_n=-4$.

Step4: Calculate the sum

$S_5=\frac{5\times(4+( - 4))}{2}=\frac{5\times0}{2}=0$.

Answer:

1. B. $f(n)=-2 + 4(n - 1)$
2. C. $0$ (assuming the option C is $0$ as it is cut off in the image)