Math CentralQuandaries & Queries


Question from alicia, a parent:
what is the next number 1,2,6,42,1806------? is it 1954?

We have four responses for you


There are always many ways to extend a sequence, some "neater" than others. Here, we note that the given numbers are growing fast, so 1954 seems unlikely.

A good trick for analyzing sequences is to look at differences of adjacent terms:

1 2 6 42 1806 Y
1 4 36 1764 X

The first three numbers there show an obvious pattern - what?

Does 1764 fit it?

So what is the next difference at X? And what goes at Y?

Alternatively, where numbers grow fast, you can try looking at ratios of adjacent terms:

1 2 6 42 1806
2 3 7 43

Good Hunting!


If you have a way to justify 1954, then nobody can tell you that you are wrong. My choice, however, would be 3263442. To get the next number in the sequence from the previous term, call it p, you multiply p*(p+1): You started with 1, so 1*(1+1)=2.
Next is 2*(2+1) = 2*3 = 6.
Then 6*7 = 42.
and 42*43 = 1806.
Thus, I would take 1806*1807.
How did you get 1954?




look at the ratios of consecutive numbers in your sequence: 2/1 = 2,6/2 = 3,42/6 =7,1806/42 = 43 and compare them to you sequence 1,2,6,42, 1806. What do you think the next ratio should be?



Hint: 1806 = 42x43.


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