What is the 34th number in this sequence 2 6 12 20?

Answer 1

The 34th number is 1190.

First number is 2 = 2n where n = 1

Second number is 4 + 2 = 2n + 2(n-1) where n = 2

Third number is 6 + 4 + 2 = 2n + 2(n-1) + 2(n -2) where n = 3

Fourth number is 8 + 6 + 4 + 2 = 2n + 2(n-1) + 2(n-2) + 2(n-3) where n = 4

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In general the nth number is 2n + 2(n-1) + 2(n-2) + 2(n-3) + ..... + 2(n-n+1) and

In general the nth number is 2n + 2(n-1) + 2(n-2) + 2(n-3) + ..... + 2(1)

Take 2 as a common factor and we get

nth number = 2(n + n -1 + n -2 + n -3 + ..... + 1) which is =

2(1 + 2 + 3 + ..... + n-3 + n-2 + n-1 + n) =

2(sum of n numbers starting with 1)

But sum of n numbers starting with 1 is (n)(n+1)/2

Hence nth number in general is (2)(n)(n+1)/2 = n(n+1)

Hence 34th number would be 34(34+1) = 34x35 = 1190