the growth does not appear exponential, yet it isn't just an easy power, either.
Note that the terms are
2*1, 3*5, 4*13, 5*25, 6*41, ...
The first factor is easy: (n+1)
The second factor is the sequence
1, 5, 13, 25, 41
1st difference: 4,8,12,16, ...
2nd difference: 4,4,4, ...
So, it is a quadratic. Some investigation shows that it is
(n-1)^2 + n^2 = 2n^2-2n+1
So, the nth term is
(n+1)(2n^2-2n+1) = 2n^3-n+1
Now, the sum of then 1st n terms will be a 4th-power polynomial.
So,
Sn = 2∑k^3 - ∑k + ∑1
k=1..n
You probably have handy the needed formulas.
if the sequence is
2,15,52,125,246....
find the nth term expression
and sum of the nth terms
1 answer