We are given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$. We are given $a_1 = 12$ and $S_5 = 90$. We need to find the common difference $d$ of the arithmetic sequence.

1. Problem Description

We are given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms denoted by $S_n$. We are given $a_1 = 12$ and $S_5 = 90$. We need to find the common difference $d$ of the arithmetic sequence.

2. Solution Steps

The formula for the sum of the first $n$ terms of an arithmetic sequence is:

$S_n = \frac{n}{2}(2a_1 + (n-1)d)$

In our case, $n=5$, $a_1 = 12$, and $S_5 = 90$. Substituting these values into the formula, we have:

$90 = \frac{5}{2}(2(12) + (5-1)d)$

$90 = \frac{5}{2}(24 + 4d)$

Multiply both sides by 2:

$180 = 5(24 + 4d)$

$180 = 120 + 20d$

Subtract 120 from both sides:

$60 = 20d$

Divide both sides by 20:

$d = \frac{60}{20}$

$d = 3$

3. Final Answer

The common difference $d$ is 3.