1. **Problem statement:** Find the sum of the arithmetic series $3 + 7 + 11 + 15 + \ldots$ to 20 terms. 2. **Formula used:** The sum of an arithmetic series is given by $$S_n = \frac{n}{2}(a_1 + a_n)$$ where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the $n$th term. 3. **Find the 20th term:** The $n$th term of an arithmetic sequence is $$a_n = a_1 + (n-1)d$$ where $d$ is the common difference. Here, $a_1 = 3$, $d = 4$, and $n = 20$. Calculate: $$a_{20} = 3 + (20-1) \times 4 = 3 + 19 \times 4 = 3 + 76 = 79$$ 4. **Calculate the sum:** $$S_{20} = \frac{20}{2} (3 + 79) = 10 \times 82 = 820$$ 5. **Check your work:** Your calculation of $a_{20} = 79$ and $S_{20} = 820$ is correct. 6. **Common mistakes to avoid:** - Make sure to use the correct common difference $d$. - Use the correct formula for the sum. - Do not confuse the number of terms $n$ with the last term index. Your work for part 1a is correct. If you had errors elsewhere, please specify which part you want to check next.