Question

Which explicit formula generates the infinite sequence 2, 9, 28, 65, 126, ...?

Answer 1

n^3+1

Explanation:

1^3=1+1=2

2^3=8+1=9

Answer 2

1. The terms of a sequence are denoted by
`u_(1) , u_(2), u_(3), u_(4), u_(5),...`

2.

`u_(1) = 2 = 2*1 u_(2) = 9 = 3*3 u_(3) = 28 = 4*7 u_(4) = 65 = 5*13 u_(5) = 126 = 6*21`

3. so it is clear that the first columns add each time by one, and the second column add by 2, then by 4, by 6, by 8 and so on.

4. consider only the second column and how we get the terms, which we will call
`t_(1) , t_(2), t_(3), t_(4), t_(5),...`:


`t_(1)=1 t_(2)=1+2 t_(3)=1+2+4=1+2+2*2 t_(4)=1+2+4+6=1+2+2*2+2*3=1+2(1+2+3)`


`t_(5)=1+2+2*2+2*3+2*4=1+2(1+2+3+4)`

5.
So


`u_(n)=(n+1)(1+2{1+2+3+....(n-1)}) =(n+1)(1+2 [(n-1)n/2]) = (n+1)(1+(n-1)n) =(n+1)( n^(2)-n+1 )`

6. We can check:
`u_(3) = (3+1)( 3^(2)-3+1 )=4*(9-3+1)=4*7=28`

7. Remark: Gauss addition formula: 1+2+3+....+n=n(n+1)/2