Multiplying, and two series in one

Learning outcome

You can continue a series built by multiplying, and you can separate one line into two series that take turns.

Concepts and reasons

If the gaps themselves grow very fast, test a short multiplication rule. The usual forms are: multiply by a fixed number, or multiply by a fixed number and then add or subtract a fixed number.

Check the rule on every step. 5 times 2 plus 1 is 11. 11 times 2 plus 1 is 23. 23 times 2 plus 1 is 47. 47 times 2 plus 1 is 95. A rule that fits only the first jump is not the rule.

Some lines are two series woven together. Look at the odd places as one series and the even places as another. 7, 10, 8, 11, 9, 12 is 7, 8, 9 in the odd places and 10, 11, 12 in the even places. The next term is the next odd-place term, which is 10.

Worked examples

1. 2, 6, 18, 54, ? Each term is times 3. Next is 162.
2. 5, 11, 23, 47, ? Each term is times 2, plus 1. Next is 95.
3. 7, 10, 8, 11, 9, 12, ? Odd places: 7, 8, 9. Even places: 10, 11, 12. Next is 10.

Common traps

Times 2 plus 1 is not the same as times 2. 47 times 2 is 94, but this rule needs 95. Do not call a series alternating until the two separated series are each regular. One tidy pair of neighbours is not enough. Cubes also grow fast. 1, 8, 27, 64 are cubes, not a multiplication by the same number.

Practice questions

1. 3, 6, 12, 24, ?
2. 4, 9, 19, 39, ?
3. 2, 9, 4, 8, 6, 7, ?

Answers and explanations

1. 48. Each term is times 2. 24 times 2 is 48.
2. 79. Each term is times 2, plus 1. 39 times 2 plus 1 is 79.
3. 8. Odd places are 2, 4, 6, so the next odd place is 8. Even places are 9, 8, 7, which continue later as 6.

Analogy

A relay team passes the baton between two runners. The odd places are one runner and the even places are the other. Read each runner own steps, not the mixed line, before you say who runs next.

Quick reference

• Fast growth: test times k, and times k plus or minus a fixed number.
• The rule must reproduce every given term.
• Split odd and even places when one rule will not fit the whole line.
• 1, 8, 27, 64 are cubes of 1, 2, 3 and 4. Next is 125.