What is an = 1*4 + 2*5 + .... +n*(n+3)?

We'll write the sum:

Sum k(k
+ 3), where k is an integer number whose values are from 1 to
n.

Sum k(k+3) = Sum (k^2 +
3k)

Sum (k^2 + 3k) = Sum k^2 + Sum
3k

Sum k^2 = 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6
(1)

Sum 3k = 3*Sum k

Sum k = 1
+ 2 + 3 + .... + n = n(n+1)/2

3*Sum k = 3n(n+1)/2
(2)

Sum k(k+3) = (1) + (2)

Sum
k(k+3) = n(n+1)(2n+1)/6 + 3n(n+1)/2

We'll factorize by
n(n+1)/2:

Sum k(k+3) = [n(n+1)/2]*[(2n+1)/3 +
3]

Sum k(k+3) = [n(n+1)/2]*[(2n + 1 +
9)/3]

Sum k(k+3) = [n(n+1)/2]*[(2n +
10)/3]

Sum k(k+3) = 2*[n(n+1)/2]*[(n +
5)/3]

We'll simplify and we'll
get:

Sum k(k+3) = [n(n+1)(n +
5)/3]

So, the value of the general term of
the string is:

 an = n(n+1)(n
+ 5)/3