The explicit formula of the sequence is -7, -3,1, 5 is [tex]$a_{n}=-7+(n-1)4$[/tex].
How to Find the Common Difference?
To find the common difference by subtracting any term in the sequence then we get,
[tex]${data-answer}amp;a_{2}-a_{1}=-3--7=4 \\[/tex]
[tex]${data-answer}amp;a_{3}-a_{2}=1--3=4 \\[/tex]
[tex]${data-answer}amp;a_{4}-a_{3}=5-1=4[/tex]
The difference in the sequence exists constant and equals the difference between two consecutive terms.
Common difference, d = 4
Estimate the sum of the sequence using the sum formula: [tex]$S u m=\frac{n\left(a_{1}+a_{n}\right)}{2}$[/tex]
[tex]$S u m=\frac{n \cdot\left(a_{1}+a_{n}\right)}{2}$[/tex]
By substituting the values, then we get
[tex]$S u m=\frac{4 \cdot\left(a_{1}+a_{n}\right)}{2}$[/tex]
[tex]$S u m=\frac{4 \cdot\left(-7+a_{n}\right)}{2}$[/tex]
[tex]$S u m=\frac{4 \cdot(-7+5)}{2}$[/tex]
Simplifying the above expression,
Sum [tex]$=\frac{4 \cdot-2}{2}$[/tex]
[tex]$S u m=-\frac{8}{2}$[/tex]
Sum = -4
The sum of this sequence exists -4.
This series corresponds to the following straight line [tex]$y=4 x \pm 7$[/tex]
The formula for expressing arithmetic sequences in their explicit form exists:
[tex]${data-answer}amp;a_{n}=a_{1}+(n-1) \cdot d[/tex]
Let the first term be [tex]$a_{1}=-7$[/tex]
common difference, d = 4
nth term be [tex]$a_{n}$[/tex]
The explicit form of this arithmetic sequence is:
[tex]$a_{n}=-7+(n-1) \cdot 4$[/tex]
Therefore, the correct answer is option (D), [tex]$a_{n}=-7+(n-1)4$[/tex].
To learn more about the explicit formula, refer:
https://brainly.com/question/11468530
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