Question

which function describes the arithmetic sequence shown? 1, 7, 13, 19, 25, 31, ... a ( f(x) = 6x + 5 ) b ( f(x) = 5x + 6 ) c ( f(x) = 5x - 6 ) d ( f(x) = 6x - 5 )

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 function \(f(x)\) (treating \(x\) as the term number, so \(x=n\)), we can also write it in slope - intercept form \(f(x)=mx + b\), where \(m\) is the common difference and we can find \(b\) using the first term.

First, find the common difference \(d\). In the sequence \(1,7,13,19,25,31,\dots\), \(d = 7-1=6\), \(13 - 7 = 6\), etc. So the slope \(m\) of the linear function (since arithmetic sequences are linear functions) is \(6\). So the function should be in the form \(f(x)=6x + b\) or \(f(x)=6x - b\) (we will check).

Step2: Find the value of \(b\)

We know that when \(x = 1\) (the first term, \(n = 1\)), \(f(1)=1\). Let's substitute into the general form \(f(x)=mx + b\) with \(m = 6\). So \(f(1)=6(1)+b\). We know \(f(1) = 1\), so:

\(1=6 + b\)

Subtract \(6\) from both sides: \(b=1 - 6=- 5\)

So the function is \(f(x)=6x-5\).

We can also check by plugging in values. For \(x = 1\): \(f(1)=6(1)-5 = 1\) (correct). For \(x = 2\): \(f(2)=6(2)-5=12 - 5 = 7\) (correct). For \(x = 3\): \(f(3)=6(3)-5 = 18 - 5=13\) (correct), which matches the sequence.

Answer:

D. \(f(x)=6x - 5\)