how to calculate n in the following sequence 1,2,4,7,11,16,22,29,37,46

Question

we are trying to work out the 2013th number in this number pattern. How do we go about doing this?

1,2,4,7,11,16,22,29,37,46,56,67

Answer 1

First write out each term so that the pattern is clearly seen:

1

1+1

1+1+2

1+1+2+3

1+1+2+3+4

1+1+2+3+4+5

...

The formula for the sum of the first n natural numbers is n(n+1)/2.

If we replace n by n-1, this becomes n(n-1)/2.

For example, the third term (n=3) is 1+1+2=4=1+(3)(2)/2.

When n=4, 1+1+2+3=7=1+(4)(3)/2.

For general n: 1+n(n-1)/2, or ½(n²-n+2).

So for n=2013, 1+2013(2012)/2=2025079.

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