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NRICH: Odds, Evens and More Evens

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions…

Here are the first few sequences from a family of related sequences:

A0=1,3,5,7,9,11,13,15,17,19,21,23,25,27,29…

A1=2,6,10,14,18,22,26,30,34,38,42…

A2=4,12,20,28,36,44,52,60…

A3=8,24,40,56,72,88,104…

A4=16,48,80,112,144…

A5=32,96,160…

A6=64…

A7=...

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Which sequences will contain the number 1000?

Once you’ve had a chance to think about it, click below to see how three different students began working on the task.

Alison started by thinking:

“I have noticed that each number is double the number in the row above. I wonder if I can work out what would go in the rows above 1000?”

Bernard started by thinking:

“I have noticed that in A1, the numbers which end in a 0 are 10, 30, 50… If I carry on going up in 20s I won’t hit 1000, so I know 1000 isn’t in A1.”

Charlie started by thinking:

“I have noticed that each number in A1 is 2 more than a multiple of 4. I know 1000 is 250×4 so it can’t be in A1.”

Can you take each of their starting ideas and develop them into a solution?

Here are some further questions you might like to consider:

How many of the numbers from 1 to 63 appear in the first sequence? The second sequence? …

Do all positive whole numbers appear in a sequence?
Do any numbers appear more than once?
Which sequence will be the longest?

Given any number, how can you work out in which sequence it belongs?
How can you describe the nth term in the sequence A0A1A2? … Am?

Age 11 to 14

Algebra & Pre-Algebra, Exponential Equations, Patterning & Sequencing
Middle School, High School, Educator

## Organization

NRICH (University of Cambridge)

Challenge