Do now Sept. 16 6 of 6 A POSSIBLE POINTS: 16.67 The first five terms of a sequence are \(a_ 1= 5, a_ 2= 7, a_ 3= 9, a_ 4= 11 \text { and } a_ 5= 13\) Based on this information , what is an equation that can be used to find the nth term . SELECT TWO \(a_ n= 3n- 5\) \(a_ n= 3+ 2( n- 1) \) \(a_ n= 5+ 2( n- 1) \) \(a_ n= 2n+ 3\) 

To find the equation for the nth term of the sequence, we observe the pattern in the given terms:
\[a_ 1 = 5, a_ 2 = 7, a_ 3 = 9, a_ 4 = 11, a_ 5 = 13\]
The sequence increases by 2 each time, which suggests a linear relationship. We can use the formula for an arithmetic sequence:
\[a_ n = a_ 1 + ( n- 1) d\]
where \(a_ 1\) is the first term and \(d\) is the common difference.
Here, \(a_ 1 = 5\) and \(d = 2\):
\[a_ n = 5 + 2( n- 1) \]
Simplifying this:
\[a_ n = 5 + 2n - 2\]
\[a_ n = 2n + 3\]
So, the two correct equations are:
\[a_ n = 5 + 2( n- 1) \]
and
\[a_ n = 2n + 3\]

Supplemental Knowledge

An arithmetic sequence is defined as an ordered string of numbers wherein each term differs by exactly one unit - known as its common difference. To find the nth term of an arithmetic sequence, you can use the formula:
\[a_ n = a_ 1 + ( n- 1) d\]
where \(a_ n\) is the nth term, \(a_ 1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.

Theory in Practice

Consider creating a weekly savings plan whereby starting with 5 dollars and increasing savings by 2 dollars every week until saving amounts form an arithmetic sequence: 5, 7, 9 11 13 etc. Using formula for arithmetic sequences allows you to predict exactly how much will have been saved at any point throughout the week.

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