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

What kind of sequence do running totals make?

Add up an arithmetic sequence a term at a time and the running totals form a sequence of their own. This lesson finds what kind it is, and how to get the terms back from the totals.

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 10: find the second difference of 1, 4, 9, 16, 25.
    Answer2.
  2. From Year 9: expand (n − 1)2.
    Answern2 − 2n + 1.
  3. From lesson 1: add 1 + 2 + 3 + 4 + 5 + 6 by pairing the ends.
    Answer21. 3 pairs of 7.
Words used in this lesson
running total
the sum of the terms so far, updated at each term. For example: S5.
partial sum
another name for the sum of the first n terms. For example: Sn.
second difference
the difference between neighbouring gaps. For example: the gaps rise by 2.
coefficient
the number multiplying a letter. For example: the 3 in 3n.
quadratic
a rule with an n squared term and no higher power. For example: n2 + 4.

Do running totals go up by the same amount?

Hadiza in Kano saves 20 naira in her first week and 5 naira more each week than the week before. She keeps a second list that shows her total savings so far at the end of each week. Her terms go up by the same amount every time.

Predict first: will the list of totals go up by the same amount each week? If not, what will be constant about it? Then use the figure to try different starting amounts and differences, and read the gaps between totals.

2
2
2second difference of the totals
42total of six terms

First term 2, difference 2: the terms are 2, 4, 6, 8, 10, 12, the running totals are 2, 6, 12, 20, 30, 42, and the totals rise by 2 more each time.

Show me the steps
  1. First term 2, difference 2: the terms are 2, 4, 6, 8, 10, 12 and the totals are 2, 6, 12, 20, 30, 42. The gaps between totals are 4, 6, 8, 10, 12, so the second difference is 2.
  2. First term 5, difference 0: every term is 5, the totals are 5, 10, 15, 20, 25, 30, and the gaps are all 5, so the second difference is 0.
  3. First term 6, difference 3: the terms are 6, 9, 12, 15, 18, 21 and the totals are 6, 15, 27, 42, 60, 81. The gaps are 9, 12, 15, 18, 21, so the second difference is 3.
  4. Any first term a and difference d: the totals have second difference d, so the totals follow (d ÷ 2)n2 + (a − d ÷ 2)n.

When d is not zero, the totals of an arithmetic sequence are a quadratic sequence. The gap between one total and the next is a term, and the terms rise by d, so the gaps rise by d: the second difference of the totals is d. That makes the n2 coefficient half of it, and

Sn = (d ÷ 2)n2 + (a − d ÷ 2)n.

The same rule comes from lesson 1: n ÷ 2 × (2a + (n − 1)d) = (d ÷ 2)n2 + (a − d ÷ 2)n. When d = 0 the totals are na, which is linear.

To go the other way, the nth term is the total of n terms minus the total of n − 1 terms: un = Sn − Sn−1, with u1 = S1.

Two common slips. Reading Sn as the nth term: Sn is the whole stack of n terms, and the nth term is only its top piece. And forgetting that the first term is S1 itself, because there is no S0 term to subtract.

Worked examples

1. The terms are 3, 7, 11, 15, … Find a rule for the total Sn.

  1. The first term is a = 3 and the difference is d = 4, so d ÷ 2 = 2 and a − d ÷ 2 = 1.
  2. Sn = 2n2 + n. Check at n = 3: 18 + 3 = 21 = 3 + 7 + 11.

2. The total of the first n terms is Sn = 3n2 + 2n. Find the nth term.

  1. Sn−1 = 3(n − 1)2 + 2(n − 1) = 3n2 − 6n + 3 + 2n − 2 = 3n2 − 4n + 1.
  2. un = Sn − Sn−1 = 3n2 + 2n − 3n2 + 4n − 1 = 6n − 1. Check: u1 = 5 = S1.

3. The totals so far are 4, 11, 21. Find the first three terms and the common difference.

  1. The terms are the gaps: 4, then 11 − 4 = 7, then 21 − 11 = 10.
  2. They go up by 3, so the terms are 4, 7, 10 and d = 3.

Your go. The first step is given. The total of the first n terms is Sn = n2 + 4n. Find the 10th term. S10 = 100 + 40 = 140.

  1. The 10th term is S10 − S9.
  2. Work out S9 and subtract.
Check the answer

S9 = 81 + 36 = 117, so the 10th term is 140 − 117 = 23.

Check before you go on

1. The total of the first n terms of a sequence is Sn = 4n2 − n. What is the 3rd term?

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. The total of the first n terms of a sequence is Sn = 2n2 + 3n. Find the 4th term.

3. Kofi in Tamale runs 7 km in the first week of his training and 2 km more each week than the week before. Use Sn = (d ÷ 2)n2 + (a − d ÷ 2)n to find how far he runs in total in the first 15 weeks, in km.

Level 2 · Building

4. The totals of the first 1, 2, 3 and 4 terms of an arithmetic sequence are 9, 22, 39 and 60. What is the common difference?

5. The total of the first n terms of an arithmetic sequence is Sn = 6n2 − n. What is the common difference?

Level 3 · Stretch

6. The total of the first n terms of a sequence is Sn = 3n2 + pn, and the 4th term is 31. Find p.

7. The total of the first n terms of a sequence is Sn = n2 + 6n. Which term of the sequence is equal to 41?

Something to think about

Running totals of an arithmetic sequence are quadratic. Running totals of a quadratic sequence should be one step further up. Guess what shape they have.

One way to think about it

Cubic. Each running total adds a term of a quadratic, so the gaps between totals are quadratic, and the totals then have a constant third difference. Each added level of summing raises the power by one.

Exit ticket

8. The total of the first n terms of a sequence is Sn = 2n2 + 5n. Which list gives the first four terms of the sequence?