College Algebra Quiz: Sum Of Arithmetic Series

What this quiz covers

This quiz focuses on Sum Of Arithmetic Series, giving you a quick way to practice the rules, question types, and explanations that matter most for College Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 144 (correct answer)
2. 156
3. 168
4. 180

Explanation: This is an arithmetic sequence with $a_1 = 23$, $d = -2$, and last term $a_n = 1$. First find $n$: $1 = 23 + (n-1)(-2)$, so $1 = 23 - 2n + 2 = 25 - 2n$, giving $2n = 24$, so $n = 12$. The sum is $S_{12} = \frac{12}{2}(23 + 1) = 6(24) = 144$. Choice B results from miscounting the number of terms (using 13 instead of 12). Choice C comes from using the wrong first term (25 instead of 23). Choice D results from using $d = -1$ instead of $d = -2$.

Question 2

1. 485 (correct answer)
2. 525
3. 565
4. 605

Explanation: Let $S_n$ denote the sum of the first $n$ terms. We have $S_{10} = 85$ and $S_{20} - S_{10} = 285$, so $S_{20} = 370$. For an arithmetic sequence, $S_n = \frac{n}{2}[2a_1 + (n-1)d]$. From $S_{10} = 85$: $\frac{10}{2}[2a_1 + 9d] = 85$, so $2a_1 + 9d = 17$. From $S_{20} = 370$: $\frac{20}{2}[2a_1 + 19d] = 370$, so $2a_1 + 19d = 37$. Subtracting: $10d = 20$, so $d = 2$. Then $2a_1 + 18 = 17$, so $a_1 = -0.5$. The sum of terms 21-30 is $S_{30} - S_{20}$. We have $S_{30} = \frac{30}{2}[2(-0.5) + 29(2)] = 15[-1 + 58] = 15 \times 57 = 855$. Therefore, $S_{30} - S_{20} = 855 - 370 = 485$. Choice B results from using $d = 2.4$ instead of $d = 2$. Choice C comes from arithmetic errors in the linear system. Choice D results from incorrectly calculating $S_{30}$.