Key takeaway: This precalculus sequences series limits test lasts 45 minutes and has eight questions for a total of 20 points. It covers sigma notation, arithmetic and geometric series, repeating decimals, the binomial theorem and limits. Show all work, and keep fractions exact unless a question asks for a decimal. Standards A-SSE.B.4, A-APR.C.5 and F-BF.A.2 shape the questions, and the final item uses a bouncing ball model.
This 45-minute test has 8 questions worth 20 points. It covers sigma notation, arithmetic and geometric series, repeating decimals, the binomial theorem and limits. Show all work, and keep fractions exact unless the question asks for a decimal.
- Grade: Precalculus
- Chapter: Series, Binomial Theorem and an Introduction to Limits
- Common Core: A-APR.C.5, A-SSE.B.4, F-BF.A.2
- Time: 45 minutes
- Total: 20 points (Question 1: 2 pts; Question 2: 2 pts; Question 3: 3 pts; Question 4: 2 pts; Question 5: 3 pts; Question 6: 2 pts; Question 7: 3 pts; Question 8: 3 pts)
Precalculus Sequences Series Limits Test: questions 1–5
1. Sigma notation (2 points)
Consider \(\sum_{k=1}^{6}(2k-3)\).
- Write out all six terms. (1 point)
- Find the sum. (1 point)
2. Arithmetic series (2 points)
An arithmetic sequence begins \(14, 11, 8, \) … .
- Find \(a_{20}\). (1 point)
- Find the sum of the first 20 terms. (1 point)
3. Geometric series (3 points)
Answer each part.
- Find the sum of the first 7 terms of \(2-6+18-\) … . (1 point)
- Find \(20+8+3.2+\) … to infinity. (1 point)
- Does \(7+8.75+10.9375+\) … have a sum? Explain. (1 point)
4. Repeating decimal (2 points)
Write \(0.\overline{45}\) as a fraction in lowest terms.
5. Binomial expansion (3 points)
Rows 0 through 5 of Pascal’s triangle are shown with four entries missing.
- Fill in the four missing entries. (1 point)
- Expand \((x-2)^{4}\). (2 points)
Precalculus Sequences Series Limits Test: questions 6–8
6. Coefficient of a term (2 points)
Find the coefficient of \(x^{4}\) in the expansion of \((x+2)^{9}\).
7. Limits (3 points)
The graph shows \(h(x)=\frac{x^{2}+x-6}{x-2}\).
- Find \(\lim_{x\to 2}h(x)\). (1 point)
- Is \(h(2)\) defined? Explain. (1 point)
- Find \(\lim_{n\to\infty}\frac{5n-2}{n+1}\). (1 point)
8. Bouncing ball (3 points)
A ball is dropped from 8 feet and rebounds to 75% of its previous height on each bounce. Find the total distance it travels before coming to rest.
How should you plan your time on this test?
Questions 1, 2 and 4 are short, so aim to finish them in about 10 minutes. Then spend 15 minutes on the geometric series in question 3 and the binomial work in questions 5 and 6. Save 20 minutes for the limits and the ball problem.
What does question 3 ask about geometric series?
You sum the first 7 terms of \(2-6+18-\cdots\), where the ratio is \(-3\). Then you find the infinite sum of \(20+8+3.2+\cdots\), where the ratio is 0.4. Check \(|r|\) first, because that decides whether any infinite sum exists.
Which precalculus sequences series limits test mistakes are common?
- Using \(n\) terms instead of \(n-1\) steps when finding \(a_{20}\) in question 2.
- Writing \(0.\overline{45}\) as \(\frac{45}{100}\) in question 4.
- Dropping the power of 2 in the coefficient of \(x^4\) in question 6.
How does question 8 use a limit idea?
A ball dropped from 8 feet rebounds to 75% of its height each time. The total path adds the first fall to twice the infinite rebound sum. Because the ratio is 0.75, the sum exists.
What is a good plan after finishing the series test?
Grade with the key, then rewrite your longest wrong solution from scratch. After that, make flash cards for the sum formulas you misused. Finally, wait two days and redo the questions you missed without looking at either the cards or the key.
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Precalculus Series, Binomial Theorem and an Introduction to Limits test
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