Grade by Grade
C. Discrete Functions · Sequences and Series

Arithmetic Series

C2.3 C2.4About 75 minutes
Learning goals
  • I can explain the difference between a sequence and a series.
  • I can develop the formula for the sum of an arithmetic series.
  • I can use Sₙ = n(a + tₙ)/2 and Sₙ = n[2a + (n − 1)d]/2 to find sums.
  • I can find the number of terms when I know the sum.

Words to know

series
The sum of the terms of a sequence, such as 3 + 7 + 11 + 15.
arithmetic series
The sum of the terms of an arithmetic sequence.
Sₙ
The sum of the first n terms: Sₙ = t₁ + t₂ + … + tₙ.
partial sum
The sum of the first n terms of a sequence; S₁, S₂, S₃, … are the partial sums.

A shortcut for adding

A well-known story, possibly exaggerated over the years, says that around 1785 a teacher in the city of Brunswick, in what is now Germany, told his class to add up all the whole numbers from 1 to 100, hoping for some quiet time. Young Carl Friedrich Gauss, who became one of the greatest mathematicians in history, wrote the answer almost at once: 5050.

His trick: pair the first number with the last, 1 + 100 = 101; the second with the second-last, 2 + 99 = 101; and so on. There are 50 pairs, each adding to 101, so the total is 50 × 101 = 5050. This lesson turns that trick into a formula that adds up any arithmetic sequence.

Sequences and series

A sequence is a list: 3, 7, 11, 15, 19. A series is a sum: 3 + 7 + 11 + 15 + 19. We write Sₙ for the sum of the first n terms:

S₁ = 3, S₂ = 3 + 7 = 10, S₃ = 3 + 7 + 11 = 21, and so on.

If the terms come from an arithmetic sequence, the sum is an arithmetic series.

5 columns6 rows1 + 2 + 3 + 4 + 5the same sum, upside downTogether: a 5 × 6 rectangle = 30 squaresSo 1 + 2 + 3 + 4 + 5 = 30 ÷ 2 = 15
Two copies of the staircase 1 + 2 + 3 + 4 + 5 fit together into a 5 by 6 rectangle. The sum is half the rectangle: 15.

Developing the formula

Write the series forwards and backwards, then add the two lines. Follow a number example and the algebra side by side.

Number example: S₅ = 3 + 7 + 11 + 15 + 19.

Algebra: Sₙ = a + (a + d) + (a + 2d) + … + tₙ

Backwards: Sₙ = tₙ + (tₙ − d) + (tₙ − 2d) + … + a

Add the two lines column by column. Every column adds to the same total, a + tₙ (in the example, 3 + 19 = 22), because one line goes up by d exactly when the other goes down by d. There are n columns, so

2Sₙ = n(a + tₙ), which gives Sₙ = n(a + tₙ) ÷ 2.

If you do not know the last term, substitute tₙ = a + (n − 1)d:

Sₙ = n[2a + (n − 1)d] ÷ 2

In words: the sum is the number of terms times the average of the first and last terms.

S₅ =S₅ =2S₅ =31922+++71522+++111122+++15722+++19322= 5 × 222S₅ = 110, so S₅ = 55. In general 2Sₙ = n(a + tₙ).
Adding 3 + 7 + 11 + 15 + 19 to itself backwards. Every column sums to 22, so twice the sum is 5 × 22 = 110.
Worked example

Example 1: Gauss's sum

Find 1 + 2 + 3 + … + 100.

  1. a = 1, tₙ = 100, n = 100.
  2. S₁₀₀ = 100(1 + 100) ÷ 2 = 100(101) ÷ 2 = 5050.

5050

Worked example

Example 2: Sum of the first n terms

Find the sum of the first 20 terms of 3 + 7 + 11 + ….

  1. a = 3, d = 4, n = 20. The last term is not given, so use the second formula.
  2. S₂₀ = 20[2(3) + (20 − 1)(4)] ÷ 2
  3. = 10[6 + 76] = 10(82) = 820.

S₂₀ = 820

Worked example

Example 3: The number of terms first

Find 5 + 8 + 11 + … + 302.

  1. You need n. Use tₙ = a + (n − 1)d: 302 = 5 + (n − 1)(3) → 297 = 3(n − 1) → n − 1 = 99 → n = 100.
  2. S₁₀₀ = 100(5 + 302) ÷ 2 = 50(307) = 15 350.

15 350

Worked example

Example 4: A negative common difference

Find the sum of the first 20 terms of 50 + 46 + 42 + ….

  1. a = 50, d = −4, n = 20.
  2. S₂₀ = 20[2(50) + 19(−4)] ÷ 2 = 10[100 − 76] = 10(24) = 240.
  3. (The terms become negative after t₁₃ = 2: t₁₄ = −2, …, t₂₀ = −26. The negative terms cancel part of the positive ones.)

S₂₀ = 240

Worked example

Example 5: Finding the number of terms from the sum

How many terms of 2 + 6 + 10 + … must be added to get 800?

  1. a = 2, d = 4. Sₙ = n[2(2) + (n − 1)(4)] ÷ 2 = n[4 + 4n − 4] ÷ 2 = n(4n) ÷ 2 = 2n².
  2. 2n² = 800 → n² = 400 → n = 20 (n must be positive).
  3. Check: t₂₀ = 2 + 19(4) = 78 and S₂₀ = 20(2 + 78) ÷ 2 = 800 ✓.

20 terms

Worked example

Example 6: Back to the concert chairs

The concert at the winter carnival has 20 rows of chairs. Row 1 has 12 chairs and each row has 2 more than the row in front. How many chairs are there altogether?

  1. a = 12, d = 2, n = 20. Last row: t₂₀ = 12 + 19(2) = 50.
  2. S₂₀ = 20(12 + 50) ÷ 2 = 10(62) = 620 chairs.

620 chairs

Just for fun

Arithmetic series would make great dance-party organizers.

They always pair the first person in line with the last, and somehow every couple scores exactly the same.

Hands-on activity

The card trick

You need: a deck of playing cards (use one suit, Ace to 10, with Ace = 1), or 10 slips of paper numbered 1 to 10.

  1. Lay the cards in a row, 1 to 10. Below them, lay a second row from 10 down to 1 (use another suit, or write the numbers on paper).
  2. Add each column. What do you notice?
  3. Use the columns to find 1 + 2 + … + 10. Check by adding normally.
  4. Remove the Ace and the 2 so the row is 3 to 10. Predict the sum using Sₙ = n(a + tₙ) ÷ 2, then check.
  5. Challenge a family member: "Name the sum of 1 to 50 before I can."

What to look for: every column adds to 11, and there are 10 columns, so twice the sum is 110 and the sum is 55. For 3 to 10: n = 8, S = 8(3 + 10) ÷ 2 = 52 (= 55 − 1 − 2 ✓). For 1 to 50: 50(51) ÷ 2 = 1275.

Your turn

1

Find 1 + 2 + 3 + … + 50.

2

Find the sum of the first 15 terms of 4 + 9 + 14 + ….

3

Find 7 + 10 + 13 + … + 100.

4

Find the sum of the first 12 terms of 30 + 25 + 20 + ….

5

An arithmetic series has a = 6, tₙ = 94 and n = 12. Find S₁₂.

6

Find the sum of all the even numbers from 2 to 200.

7

How many terms of 1 + 3 + 5 + … add up to 225?

More practice: warm-up

1

What is the difference between a sequence and a series?

2

For 2, 5, 8, 11, find S₁, S₂, S₃ and S₄.

3

Find 10 + 20 + 30 + … + 100.

4

An arithmetic series has 8 terms, first term 3 and last term 24. Find the sum.

5

Write the formula for Sₙ when you know a, d and n.

More practice: on your own

1

Find −12 − 8 − 4 + 0 + … + 48.

2

Find the sum of the first 40 terms of 1/2 + 1 + 3/2 + ….

3

Find the sum of all the multiples of 3 between 1 and 300.

4

In an arithmetic series S₁₀ = 155 and a = 2. Find d.

5

How many terms of 3 + 7 + 11 + … add up to 210?

Challenge

1

The sum of the first n terms of a series is Sₙ = 3n² + 2n. Find t₁, t₂ and the general term. Is the sequence arithmetic?

2

Explain with a picture or algebra why 1 + 3 + 5 + … + (2n − 1) = n².

3

A ski hill's chairlift line has 1 person in the front row and each row behind has 1 more person, up to some row. Then a second group arrives so the line keeps the same pattern. If the line had 210 people in total, how many rows were there? If the second group was 155 people, how many rows did the first group fill?

Check your understanding

Quiz

1.

Which is a series?

2.

What is 1 + 2 + 3 + … + 40?

3.

Find the sum of the first 10 terms of 5 + 9 + 13 + ….

4.

An arithmetic series has 25 terms, first term 4 and last term 76. What is its sum?

5.

How many terms are in 6 + 11 + 16 + … + 101?

6.

Why does the formula Sₙ = n(a + tₙ) ÷ 2 work?

For parents and teachers

Big idea: Expectation C2.3 asks students to develop the formula for the sum of an arithmetic series through investigation (diagrams, patterning, connecting a number example to the algebra) and use it to find the sum of a given number of consecutive terms. C2.4 adds problems.

How to help: Try the Gauss pairing trick together with small numbers, then ask your teen to explain why every pair has the same total. The "average of first and last, times the number of terms" version is easy to remember.

Common difficulties:

  • Using the last term's value instead of the number of terms for n. Find n first (Example 3).
  • Forgetting to divide by 2.
  • Sign errors when d is negative (Example 4).
  • When solving for n gives a quadratic, keeping a negative or non-whole answer. n must be a natural number.