Answer
This is an arithmetic sequence.
The common difference $d$ is $15$.
$a_32 = 468$
Work Step by Step
Let's look at the terms to see if we can find a pattern:
$a _2 - a_1 = 18 - 3 = 15$
$a_3 - a_2 = 33 - 18 = 15$
$a_4 - a_3 = 48 - 33= 15$
This is an arithmetic sequence because the same number is added to the previous term to get the next term, which means there is a common difference, and that difference $d$ is $15$.
We can find the 32nd term by figuring out the explicit formula of a sequence, which will give us the exact term we are looking for.
Use the explicit formula for arithmetic sequences, which is $a_n = a_1 + (n - 1)d$, where $a_1$ is the first term of the sequence and $d$ is the common difference; in this exercise, $a_1$ is $3$ and $d$ is $15$:
$a_n = 3 + (n - 1)(15)$
Substitute $32$ for $n$ to find the $32$nd term of the sequence:
$a_32 = 3 + (32 - 1)(15)$
Evaluate what's in parentheses first:
$a_32 = 3 + (31)(15)$
Then multiply:
$a_32 = 3 + 465$
Add to solve:
$a_32 = 468$
