Question

select the function that correctly models the arithmetic sequence {16,21,26,31,36}. (1 point)
\\( y = \frac{1}{5}x + 11 \\)
\\( y = 5x \\)
\\( y = 11x + 5 \\)
\\( y = 5x + 11 \\)

Explanation:

Step1: Recall arithmetic sequence formula

The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference. For a linear function \(y = mx + b\) (where \(x\) represents the term number and \(y\) represents the term value), the slope \(m\) is the common difference \(d\).
First, find the common difference \(d\) of the sequence \(\{16,21,26,31,36\}\). \(d=21 - 16=5\), \(d = 26-21 = 5\), so \(m = 5\).

Step2: Find the y - intercept \(b\)

We can use the first term. When \(x = 1\) (the first term, term number 1), \(y=16\). Substitute into \(y=mx + b\) with \(m = 5\):
\(16=5\times1 + b\)
\(16=5 + b\)
Subtract 5 from both sides: \(b=16 - 5=11\).

So the function is \(y = 5x+11\). We can also check with other terms. For \(x = 2\), \(y=5\times2+11=10 + 11 = 21\) (matches the second term). For \(x = 3\), \(y=5\times3+11=15 + 11 = 26\) (matches the third term), etc.

Answer:

\(y = 5x + 11\) (the option: \(y=5x + 11\))