Year 11 Extended · Unit 3 Sequences, series, formulae and proportion · Lesson 2 of 7

How can a long sum be written in one short line?

Mathematicians write a sum with the Greek letter sigma, a rule and two limits. This lesson reads that line as a list of terms, counts the terms, and uses it to split a sum into easier ones.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 9: evaluate 3 × 22 − 5.
    Answer7.
  2. From Year 7: how many whole numbers are there from 4 to 15, counting both ends?
    Answer12.
  3. From lesson 1: add 3 + 5 + 7 + 9.
    Answer24.
Words used in this lesson
sigma
the symbol used for adding up the terms of a rule. For example: Σ.
limits
the first and last values of the counter k. For example: 1 and 6.
index
the counter letter, such as k, that runs through the values. For example: k.
term
one value of the rule for one value of k. For example: the 3rd one.
evaluate
work out the number a sum equals. For example: add them.

What does one line of symbols say?

Amina in Lagos saves a little more each week: 3 naira, then 5, 7, 9 and 11. She wants to write her total savings so that anyone, or a spreadsheet, can read it without listing the numbers. A mathematician writes it as Σ with the start below, the end above, and the rule beside it:  Σk=15 (2k + 1).

Predict first: what do you think the k = 1 below and the 5 above are telling you to do? Then use the figure to try different rules and different starting values, and check the terms and the count.

1
1
4
4number of terms
26sum of the terms

Rule 3k − 1, k from 1 to 4 (4 terms): the terms are 2, 5, 8, 11 and their sum is 26.

Show me the steps
  1. Rule 3k − 1, k from 1 to 4: the terms are 2, 5, 8, 11, there are 4 terms and the sum is 26.
  2. Rule k squared, k from 2 to 4: the terms are 4, 9, 16, there are 3 terms and the sum is 29. Starting at 2 leaves out the term for k = 1.
  3. Rule 2 to the power k, k from 1 to 6: the terms are 2, 4, 8, 16, 32, 64, there are 6 terms and the sum is 126.
  4. Any rule, k from a to b: the number of terms is b − a + 1, and the sum is the rule's value at each k, added.

Sigma means “add up”. The sum from k = a to k = b of a rule means: put k = a, then a + 1, and so on up to b, into the rule, and add the results. There are b − a + 1 terms, because both ends are included.

Three facts save work: the sum of a rule plus another splits into two sums; a constant multiplier comes outside the sum; and adding a constant n times gives n times the constant. Each comes from writing the sum out: for k = 1 to 3, (3k + 1) gives (3 + 1) + (6 + 1) + (9 + 1) = 3 × (1 + 2 + 3) + 3 × 1. The letter k is only a counter, so any letter would do.

Two common slips. Counting the terms as b − a: the sum from k = 10 to k = 15 has 15 − 10 + 1 = 6 terms, not 5. And treating a constant as a single term: the sum from k = 1 to 5 of 4 is 4 added five times, which is 20, not 4.

Worked examples

1. Write out and evaluate the sum from k = 1 to 4 of (k2 + 1).

  1. Put k = 1, 2, 3, 4 into the rule: 2, 5, 10, 17.
  2. Add them: 2 + 5 + 10 + 17 = 34.

2. Write 4 + 7 + 10 + 13 + 16 + 19 using sigma, and find its sum.

  1. The terms go up by 3 and the first is 4, which is 3 × 1 + 1. The rule is 3k + 1 and k runs from 1 to 6, so the sum is the sum from k = 1 to 6 of (3k + 1).
  2. The terms are an arithmetic sequence: 6 ÷ 2 × (4 + 19) = 69.

3. Use the facts to find the sum from k = 1 to 10 of (5k − 2).

  1. Split it: 5 times the sum of k, minus 2 added ten times.
  2. The sum of k from 1 to 10 is 10 ÷ 2 × 11 = 55, so the total is 5 × 55 − 2 × 10 = 255.

Your go. The first step is given. Find the sum from k = 1 to 8 of (4k + 3). The sum of k from 1 to 8 is 36.

  1. Split it: 4 times the sum of k, plus 3 added eight times.
  2. Put in the sum of k and finish.
Check the answer

4 × 36 + 3 × 8 = 144 + 24 = 168.

Check before you go on

1. What is the value of the sum from k = 3 to k = 9 of 2?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. Evaluate the sum from k = 1 to k = 5 of (3k2 − 4).

3. Evaluate the sum from k = 4 to k = 9 of (2k + 5).

Level 2 · Building

4. A tailor in Dhaka sews 6k − 5 shirts in week k, so 1 shirt in week 1 and 7 in week 2. The sum from k = 1 to k = 20 of k is 210. Use this to find how many shirts she sews in the first 20 weeks.

5. Evaluate the sum from k = 7 to k = 23 of 3k.

Level 3 · Stretch

6. The sum from k = 1 to k = n of (2k − 1) is 144. Find n.

7. The sum from k = 1 to k = 5 of (pk + 3) is 75. Find p.

Something to think about

Does it matter which letter sits under the sigma? Is the sum from k = 1 to 4 of (2k + 1) different from the sum from j = 1 to 4 of (2j + 1)?

One way to think about it

No. The letter is only a counter that is used up inside the sum, so any letter gives the same list of terms. What matters is the rule, the first value and the last value.

Exit ticket

8. Which list shows the terms of the sum from k = 2 to k = 6 of (5 − k)?