Question

an arithmetic series consists of consecutive integers that are multiples of 4. whats the sum of the first nine terms of this sequence if the first term is 0?
a) 198
b) 160
c) 162
d) 144
question 8 (5 points)
which of the following is an arithmetic sequence with common difference - 5?
a) 8, 3, -2, -7, ...
b) 0, 5, 10, 15, ...
c) 5, -25, 125, -625, ...

Explanation:

Step1: Identify the arithmetic - series formula

The sum of the first $n$ terms of an arithmetic series is given by $S_n=\frac{n(a_1 + a_n)}{2}$, where $n$ is the number of terms, $a_1$ is the first - term, and $a_n$ is the $n$th term. The formula for the $n$th term of an arithmetic sequence is $a_n=a_1+(n - 1)d$, where $d$ is the common difference.

Step2: Determine the common difference

Since the arithmetic series consists of consecutive integers that are multiples of 4 and the first term $a_1 = 0$, the common difference $d = 4$.

Step3: Find the 9th term

Using the formula $a_n=a_1+(n - 1)d$, with $n = 9$, $a_1 = 0$, and $d = 4$, we have $a_9=0+(9 - 1)\times4=32$.

Step4: Calculate the sum of the first 9 terms

Using the sum formula $S_n=\frac{n(a_1 + a_n)}{2}$, with $n = 9$, $a_1 = 0$, and $a_9 = 32$, we get $S_9=\frac{9\times(0 + 32)}{2}=\frac{9\times32}{2}=9\times16 = 144$.

for the second question:

Step1: Recall the definition of an arithmetic sequence

An arithmetic sequence has a common difference $d$ such that $a_{n+1}-a_n=d$ for all $n$.

Step2: Check each option

For option A:
$a_1 = 8$, $a_2 = 3$, $a_2-a_1=3 - 8=-5$; $a_3=-2$, $a_3 - a_2=-2 - 3=-5$; $a_4=-7$, $a_4 - a_3=-7+2=-5$. The common difference is $-5$.
For option B:
$a_1 = 0$, $a_2 = 5$, $a_2 - a_1=5-0 = 5$, the common difference is 5, not - 5.
For option C:
$a_1 = 5$, $a_2=-25$, $\frac{a_2}{a_1}=\frac{-25}{5}=-5$, this is a geometric sequence (not an arithmetic sequence) since the ratio between consecutive terms is constant, not the difference.

Answer:

D. 144