Question

what is the sum of 1+5+14+30+----- in arithmetic progress

Answer 1

I think it's 40.... im not sure

Answer 2

Notice that the sequence of differences between terms follows the pattern 4, 9, 16, i.e. 22, 32, 42.

Assuming the pattern continues, the next couple of terms would be 55 and 91, being 30+52 and 55+62

Write down the sequence of the first few sums:

1,6,20,50,105,196

Write down the sequence of differences between consecutive terms:

5,14,30,55,91

Write down the sequence of differences of that sequence:

9,16,25,36

Write down the sequence of differences of that sequence:
7,9,11
Write down the sequence of differences of that sequence:
2,2
Having reached a constant sequence, we can use the initial terms of each of these sequences as coefficients of an expression for the nth term of the sequence of sums:

sn=10!+51!(n−1)+92!(n−1)(n−2)+73!(n−1)(n−2)(n−3)+24!(n−1)(n−2)(n−3)(n−4)

sn=1+5(n−1)+92(n2−3n+2)+76(n3−6n2+11n−6)+112(n4−10n3+35n2−50n+24)

sn=112(n4+4n3+5n2+2n)