series problem?

Question

here is a question
http://docs.google.com/Doc?id=dc83hzcs_212fgq97g

i tried to solve but it does not match with the answers given :(

Answer 1

1277 is the answer.
You are correct in that the differences increase by 1, since:
a(n+2) + a(n) = 2a(n+1) + 1
a(n+2) - a(n+1) = a(n+1) - a(n) + 1
But you need to assume a general term of an^2 + bn + c
and put in three values to get 3 equations with 3 unknowns.
This yields a = 1/2, b = -1/2, c = 2; or
a(n) = (n^2 - n + 4)/2
a(51) = 1277

Answer 2

The general term should have the form a_n = bn^2 + cn + d
You should get b = 1/2, c = -1/2, d = 2. Then a_51 is 1277.

Answer 3

Approach the series with a calm eye. This will allow you to determine its location to within a tolerance of 1 micron. Such an expression is meaningless, however, askance of the philosophical and metaphysical ramifications of such an event. I would suggest creating a computer program that outputs random computer programs. Simply run this program until it outputs a program that solves your current program.