Precalculus Sequences, Series and Limits Worksheet – 25 Problems PDF

Precalculus Sequences Series Limits Worksheet: problems 1–5

1. Evaluate a sum

Evaluate \(\sum_{k=1}^{5}(3k-1)\).

2. Sums of powers

Evaluate each sum.

1. \(\sum_{k=1}^{4}k^{2}\)
2. \(\sum_{k=0}^{3}2^{k}\)

3. Write in sigma notation

Write each sum in sigma notation.

1. 3 + 6 + 9 + 12 + 15
2. 2 + 4 + 8 + 16 + 32

4. Explicit and recursive rules

Write an explicit rule and a recursive rule for each sequence, then find the requested term.

1. 5, 9, 13, 17, … ; find \(a_{20}\)
2. 2, 6, 18, 54, … ; find \(a_{6}\)

5. Find the nth term

An arithmetic sequence has \(a_{1}=7\) and common difference \(d=4\).

1. Find \(a_{20}\).
2. Which term of the sequence equals 103?

Precalculus Sequences Series Limits Worksheet: problems 6–10

6. Arithmetic series

Find the sum of the first 15 terms of the arithmetic series \(12+19+26+\) … .

7. Decreasing series

Find \(100+97+94+\) … \(+4+1\).

8. Finite geometric series

Find each sum.

1. \(3+6+12+\) … , first 8 terms
2. \(81+27+9+3+1\) using the geometric series formula

9. Does it converge?

Decide whether each infinite geometric series converges. If it does, find its sum.

1. 8 + 4 + 2 + 1 + …
2. 5 + 10 + 20 + 40 + …
3. 9 − 3 + 1 − 1/3 + …
4. 2 + 2.2 + 2.42 + 2.662 + …

10. Infinite geometric sum

Find each sum.

1. \(12+4+\frac{4}{3}+\frac{4}{9}+\) …
2. \(\sum_{k=0}^{\infty}5(0.2)^{k}\)

Precalculus Sequences Series Limits Worksheet: problems 11–15

11. Repeating decimals

Write each repeating decimal as a fraction in lowest terms using an infinite geometric series.

1. \(0.\overline{27}\)
2. \(0.\overline{6}\)

12. Pascal's triangle

The triangle shows rows 0 through 6 with three entries missing.

1. Find the three missing entries.
2. Write row 7.

13. Expand a binomial

Expand each binomial.

1. \((x+2)^{5}\)
2. \((x-3)^{3}\)

14. Binomial with a coefficient

Expand \((2x-1)^{4}\).

15. Find a specific term

Use the binomial theorem.

1. Find the coefficient of \(x^{3}\) in \((x+3)^{7}\).
2. Find the term containing \(x^{2}\) in \((2x+1)^{6}\).

Precalculus Sequences Series Limits Worksheet: problems 16–20

16. Limit of a sequence

Let \(a_{n}=\frac{3n+1}{n+2}\).

1. Find \(a_{10}\), \(a_{100}\) and \(a_{1000}\) to 4 decimal places.
2. What is \(\lim_{n\to\infty}a_{n}\)?

17. Limit from a graph

The graph shows \(f(x)=\frac{x^{2}-9}{x-3}\).

1. Is \(f(3)\) defined?
2. Find \(\lim_{x\to 3}f(x)\).

18. Limit by factoring

Find each limit.

1. \(\lim_{x\to 3}\frac{x^{2}+x-12}{x-3}\)
2. \(\lim_{x\to 5}\frac{x^{2}-25}{x-5}\)

19. When does it converge?

Consider \(\sum_{k=0}^{\infty}(\frac{x}{3})^{k}\).

1. For which values of x does the series converge?
2. Find the sum when \(x=1\).
3. Find the sum when \(x=-\frac{3}{2}\).

20. Theater seats

The bar chart shows the number of seats in rows 1 to 6 of the school auditorium, and the pattern continues. Find the total number of seats in rows 1 through 6, then in rows 1 through 20.

Precalculus Sequences Series Limits Worksheet: problems 21–25

21. Doubling savings

Mei saves 1 cent on day 1, 2 cents on day 2, 4 cents on day 3, and keeps doubling for 12 days. Find the total she saves, in dollars.

22. Salary raises

Jamal starts a job at $40,000 a year and gets a $2,500 raise every year. Write a recursive rule for his salary, then find his salary in year 10 and his total earnings over the first 10 years.

23. Bouncing ball

A ball is dropped from 10 feet and rebounds to 60% of its previous height on every bounce, as sketched. Find the total distance the ball travels before it comes to rest.

24. Medicine in the body

Ethan takes a 100 mg dose every morning. Overnight his body removes 80% of the drug, so 20% remains. Find the amount in his body just after the dose on day 3, and the long-run amount just after a dose.

25. Cleaning a pond

A filter removes 6 pounds of algae the first week, and each week after that it removes one third as much as the week before. The graph shows the total removed after n weeks. Estimate the long-run total from the graph, then find it exactly.

What do the first ten problems build?

Problem 1 asks for \(\sum_{k=1}^{5}(3k-1)\), which equals 40. Next, a countdown from 100 to 1 in steps of 3 appears in problem 7, so you must find the number of terms before you add. Finally, two infinite series meet in problem 9: the one with ratio \(\frac{1}{2}\) converges, while the one starting \(5+10+\cdots\) does not.

How do you expand the binomials in problems 12 to 15?

First, problem 12 fills missing entries of Pascal’s triangle. Then \((2x-1)^4\) in problem 14 needs care, because each power of 2 and each minus sign matters. After that, problem 15 asks for the coefficient of \(x^3\) in \((x+3)^7\), and that coefficient uses \(\binom\,{7}{3}\cdot3^4\).

Which word problems use geometric series?

Savings double for 12 days in problem 21, starting at 1 cent. In problem 23, a ball drops from 10 feet with a 60% rebound, so add the falls and the rises separately. Likewise, problem 24 follows a 100 mg daily dose, and problem 25 removes algae at one third of the previous rate.

How does the precalculus sequences series limits worksheet introduce limits?

Problem 16 tabulates \(a_n=\frac{3n+1}{n+2}\) for \(n=10, 100, 1000\). The values climb toward 3. Next, problems 17 and 18 ask you to factor a numerator to cancel a zero, which explains why a hole in a graph still has a limit.

How can you split the 25 problems over a week?

Do problems 1 to 8 on day one and 9 to 15 on day two. Leave the limits and applications for later days. Meanwhile, mark each slow problem with a star. Review only the starred ones before you open the answer key.

See every chapter on the Precalculus math page.