Check whether 301 is a term of the list of numbers $5, 11, 17, 23, \ldots$

Step 1: Identify the type of sequence and its properties

To determine if 301 is a term, we first need to identify if the given list of numbers is an Arithmetic Progression (AP). We do this by checking if the difference between consecutive terms is constant. In this case, the common difference $d$ is 6, which indicates it is an AP. The first term $a$ is 5.

Step 2: Set up the formula for the $n$ -th term

The formula for the $n$ -th term of an Arithmetic Progression is $a_n = a + (n-1)d$, where $a_n$ is the $n$ -th term, $a$ is the first term, $n$ is the term number, and $d$ is the common difference.

Step 3: Substitute known values into the formula

We assume that 301 is the $n$ -th term ($a_n = 301$) and substitute the values of the first term ($a = 5$) and the common difference ($d = 6$) into the formula. This equation will allow us to solve for $n$.

Step 4: Solve for $n$

Now we simplify the equation and solve for $n$. We distribute the 6, combine like terms, and then isolate $n$ by performing arithmetic operations.

Step 5: Check if $n$ is a positive integer

For 301 to be a term in the sequence, $n$ must be a positive integer. Since $n = \frac{151}{3}$ is not an integer (it's a fraction), 301 is not a term of the given Arithmetic Progression. The $n$ -th term in an AP must correspond to a whole number position in the sequence.

More from this chapter

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