Given the table:
x y
1 2
2 0
3 -2
4 -4
Difference between x values:
4 - 3 = 1
3 - 2 = 1
2 - 1 = 1
Difference between y values:
-4 - - 2 = -4 + 2 = -2
-2 - - 0 = - 2 + 0 = -2
0 - 2 = -2
This table represents an Arithmetic sequence.
An arithmetic sequence is a sequence that has a constant difference between the terms.
From the table given, the difference between the consecutive terms is constant.
We have:
First term = 2
Second term = 0
Third term = -2
Fourth term = -4
Arithmetic sequence has a common difference.
The common difference = -2
Use the arithmetic sequence formula:
[tex]a_n=a_1+(n-1)d[/tex]Where:
a1 = first term
n = number of terms
d = common difference
To write the recursive formula, we have:
[tex]\begin{cases}a_1=2 \\ a_n=a_{n-1}_{}-2\end{cases}[/tex]To write the explicit formula, we have:
[tex]a_n=2_{}+(n-1)(-2)[/tex]Let's simplify the explicit formula:
[tex]\begin{gathered} a_n=2+(n-1)(-2) \\ \\ a_n=2+\text{ n(-2) -1(-2)} \\ \\ a_n=2-2n+2 \\ \\ a_n=-2n+2+2 \\ \\ a_n=-2n+4 \end{gathered}[/tex]To find the 10th term, let's use the explicit formula.
Substitute 10 for n
[tex]\begin{gathered} a_{10}=-2(10)+4 \\ \\ a_{10}=-20+4 \\ \\ a_{10}=-16 \end{gathered}[/tex]Therefore, the 10th term of the arithmetic sequence is -16
ANSWER:
Arithmetic Sequence
Common difference
Recursive formula:
[tex]\begin{cases}a_1=2 \\ a_n=a_{n-1}-2\end{cases}[/tex]Explicit formula:
[tex]a_n=-2n+4[/tex]