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Predicting What Comes Next: Exploring Sequences and Progressions

Adding the first n

A hall has 20 rows. Row 1 has 1 chair, row 2 has 2, and so on. How many chairs?

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The list 1, 2, 3, 4, drawn as a rising staircase.
Step 1 of 4
  1. The list 1, 2, 3, 4, drawn as a rising staircase.
  2. A mirror copy fills every column to height 5.
  3. Two copies: 4 columns × 5 = 20.
  4. One copy: 20 ÷ 2 = 10.

The rule

The hall has 210 chairs. The sum of a list is its total.

Write the list twice, the second time backwards. Add each number to the one below it.

1 + 2 + 3 + 4 and 4 + 3 + 2 + 1. The pairs are 1 + 4, 2 + 3, 3 + 2 and 4 + 1. Each pair makes 5. There are 4 pairs.

Two copies make 4 × 5 = 20. One copy is half of that: 10.

Now take the first n numbers, 1 to n. Each pair makes n + 1, and there are n pairs. So one copy is half of n × (n + 1).

1 + 2 + … + n = n(n + 1) ÷ 2

For the hall, n = 20. 20 × 21 = 420, and 420 ÷ 2 = 210 chairs.

The sums 1, 3, 6, 10, 15, … are the triangular numbers. The nth one is n(n + 1) ÷ 2. Rows of 1, 2, 3, … dots make a triangle.

The rule, checked at three sizes
nRectangleSum
44 × 5 = 2010
2020 × 21 = 420210
100100 × 101 = 101005050

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This track

  1. 0 Start here
  2. 1 A rule, not a guess
  3. 2 A rule that looks back
  4. 3 A rule that jumps straight there
  5. 4 The same step every time
  6. 5 Adding the first n
  7. 6 Checkpoint: rules, jumps and the fold
  8. 7 The same multiplier every time
  9. 8 A shape that multiplies
  10. 9 A tower and a doubling
  11. 10 Does the rule really fit?
  12. 11 Checkpoint: multipliers, stages, towers and rivals
  13. 12 The big picture
  14. 13 Track quiz

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