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C. Discrete Functions · Sequences and Series

Sequences and Series: Unit Review

C1.1 C1.2 C1.3 C1.4 C1.5 C1.6 C2.1 C2.2 C2.3 C2.4About 40 minutes
Learning goals
  • I can represent sequences in several ways and move between them.
  • I can use Pascal's triangle to expand binomials.
  • I can find terms and sums of arithmetic and geometric sequences and series.
  • I can solve real problems with sequences and series.

Words to know

sequence / series
A list of terms / the sum of the terms.
recursion formula
Gives the first term(s) and a rule for each term from earlier terms.
arithmetic
Constant difference d: tₙ = a + (n − 1)d, Sₙ = n(a + tₙ) ÷ 2.
geometric
Constant ratio r: tₙ = arⁿ⁻¹, Sₙ = a(rⁿ − 1) ÷ (r − 1).
binomial expansion
(a + b)ⁿ multiplied out; its coefficients are row n of Pascal's triangle.

Let's review!

In this unit you learned that a sequence is a discrete function on the natural numbers, and you represented sequences with general terms, function notation, recursion formulas, tables and graphs. You explored the Fibonacci numbers and Pascal's triangle, and used the triangle to expand binomials. Then you met the two great families, arithmetic and geometric sequences, developed formulas for their general terms and for the sums of their series, and used them to solve real problems.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do two or three sections at a time. Before using any formula, write down a, d or r, and n. Ask yourself: "Is this one term (tₙ) or a total (Sₙ)?" Check answers by substituting, or by writing out the first few terms. For every question you get wrong, write down why (wrong type of sequence? n off by one? sign or bracket error? term instead of sum?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Sequencestₙ = term in position nrecursion: t₁ given, tₙ from tₙ₋₁discrete function, domain ℕFibonacci: tₙ = tₙ₋₁ + tₙ₋₂Pascal's triangletₙ,ᵣ = tₙ₋₁,ᵣ₋₁ + tₙ₋₁,ᵣrow sums 2ⁿ(a + b)ⁿ: coefficients = row na powers down, b powers upArithmetic (add d)tₙ = a + (n − 1)dSₙ = n(a + tₙ) ÷ 2Sₙ = n[2a + (n − 1)d] ÷ 2graph: points on a lineGeometric (multiply by r)tₙ = arⁿ⁻¹Sₙ = a(rⁿ − 1) ÷ (r − 1), r ≠ 1Sₙ = a(1 − rⁿ) ÷ (1 − r)graph: points on an exponential curve
The whole unit on one card: sequences and recursion, Pascal's triangle, and the formulas for arithmetic and geometric sequences and series.
Just for fun

You now know enough to add the numbers 1 to 1000 in your head (500 500), count the rice on a chessboard (2⁶⁴ − 1 grains), and expand (x + 2)⁵ without breaking a sweat. Not bad for a unit that started with chairs at a winter carnival.

Sequences as Discrete Functions

1

Write the first four terms of tₙ = 6n − 5 and find t₂₀.

2

Is 77 a term of tₙ = 4n + 3? If so, which one?

3

Is the number of snowplows a city sends out each day a discrete or a continuous function of the day? Explain.

Recursive Procedures: Building a Sequence Step by Step

1

Write the first five terms of t₁ = 2, tₙ = 3tₙ₋₁ − 1.

2

Write a recursion formula for 6, 11, 16, 21, ….

3

Write a recursion formula for 1, 2, 4, 7, 11, 16, ….

Three Ways to Represent a Sequence

1

Write 5, 25, 125, … as a recursion formula and in function notation.

2

Find an explicit formula for 0, 3, 8, 15, 24, ….

3

Write a recursion formula for tₙ = 3n² .

The Fibonacci Sequence and Related Sequences

1

Write the first 10 Fibonacci numbers.

2

Find the sum of the first 10 Fibonacci numbers using a pattern.

3

A Fibonacci-type sequence has t₁ = 4 and t₂ = 1. Find t₆.

Pascal's Triangle: A Recursive Pattern

1

Write row 6 of Pascal's triangle.

2

What is the sum of row 11?

3

How many shortest routes go 3 blocks east and 3 blocks south?

Pascal's Triangle and the Binomial Expansion

1

Expand (x + 3)⁴.

2

Expand (2x − 1)³.

3

Find the 3rd term of (a − 2b)⁵.

Arithmetic Sequences and the General Term

1

Find t₂₅ for 8, 13, 18, ….

2

How many terms are in 11, 7, 3, …, −85?

3

In an arithmetic sequence t₅ = 23 and t₁₂ = 58. Find tₙ.

Geometric Sequences and the General Term

1

Find t₇ for 2, −6, 18, ….

2

How many terms are in 5, 10, 20, …, 2560?

3

In a geometric sequence t₂ = 10 and t₄ = 250. Find the possible values of r and a.

Arithmetic, Geometric or Neither?

1

Classify 3, 6, 11, 18, ….

2

Classify tₙ = 8(0.5)ⁿ.

3

Classify t₁ = 4, tₙ = tₙ₋₁ − 2.5.

Arithmetic Series

1

Find 1 + 2 + … + 200.

2

Find the sum of the first 30 terms of 2 + 5 + 8 + ….

3

Find 100 + 96 + 92 + … + 4.

Geometric Series

1

Find the sum of the first 8 terms of 1 + 3 + 9 + ….

2

Find the sum of the first 6 terms of 96 + 48 + 24 + ….

3

Find 3 − 6 + 12 − … + 768.

Solving Problems with Arithmetic and Geometric Sequences

1

A ball is dropped from 5 m and rebounds to 70% of its height each time. How high is the 3rd bounce?

2

A town of 12 000 grows by 2% each year. Estimate the population after 8 years.

3

A theatre's first row has 20 seats and each row has 2 more. Which row has 50 seats?

Solving Problems with Arithmetic and Geometric Series

1

The theatre in the last question has 16 rows. How many seats in total?

2

A ball dropped from 5 m rebounds to 70% each time. Total distance when it hits the ground the 4th time?

3

A runner runs 3 km on day 1 and adds 0.5 km each day. On which day does her total first pass 100 km?

0123456710203040506070nvaluetₙ = 2ⁿ (geometric)tₙ = 5n + 2 (arithmetic)Arithmetic: straight line of points.Geometric: curve that bends up.
Arithmetic sequences grow by equal steps (points on a line); geometric sequences with r > 1 grow by equal factors (points on an upward-bending curve), and eventually pass any arithmetic sequence.
Check your understanding

Quiz

1.

What is the domain of a sequence?

2.

For t₁ = 3, tₙ = 2tₙ₋₁ + 1, what is t₄?

3.

What is the next Fibonacci number after 21 and 34?

4.

What is the coefficient of x²y² in (x + y)⁴?

5.

What is t₁₅ of the arithmetic sequence 9, 6, 3, …?

6.

What is t₆ of the geometric sequence 2, 6, 18, …?

7.

The sequence 2, 6, 12, 20, … is:

8.

What is 2 + 7 + 12 + … + 97?

9.

What is the sum of the first 7 terms of 1 + 2 + 4 + …?

10.

A ball dropped from 6 m rebounds to 50% each time. How far has it travelled when it hits the ground the 3rd time?

For parents and teachers

Use the review to find what to revisit:

  • Mixing up position n and the term tₙ, or joining the dots on a sequence graph → Lesson 1 (Sequences as Discrete Functions).
  • Recursion formulas missing a first term, or reading tₙ₋₁ as "tₙ minus 1" → Lesson 2 (Recursive Procedures).
  • Trouble finding an explicit formula or converting between forms → Lesson 3 (Three Ways to Represent a Sequence).
  • Fibonacci patterns and two-term recursions → Lesson 4 (The Fibonacci Sequence).
  • Building rows, row and position numbering, route counting → Lesson 5 (Pascal's Triangle).
  • Expansion errors: missing brackets, lost negative signs, wrong term → Lesson 6 (Binomial Expansion).
  • Using n instead of n − 1, or wrong sign of d → Lesson 7 (Arithmetic Sequences).
  • rⁿ instead of rⁿ⁻¹, calculator errors with negative r, forgetting r = ±… → Lesson 8 (Geometric Sequences).
  • Calling a sequence arithmetic or geometric from too few terms → Lesson 9 (Arithmetic, Geometric or Neither?).
  • Arithmetic sums: finding n, dividing by 2 → Lesson 10 (Arithmetic Series).
  • Geometric sums: which form of the formula, r = 1, fractions → Lesson 11 (Geometric Series).
  • Real problems: r = 0.85 for a 15% loss, starting at time 0 → Lesson 12 (Problems with Sequences).
  • Term versus total, bouncing-ball distance → Lesson 13 (Problems with Series).

Next in this course, the unit Financial Applications uses geometric sequences and series for compound interest and annuities. In Grade 12 Data Management (MDM4U), Pascal's triangle returns when you study combinations.