IAmDumbAtMath

Sum of arithmetic sequence

 Sum32 =  n/2 ( 2a1 + (n-1)d)

             = 32/2 (2a1  + (32-1)d)

1448    = 16 (2a1 + 31d)

5th term = 18 = a1 + 4d    so   a1 = 18- 4d     use this value of a1 in the red equation above to calculate 'd'

then  ninth term will be:    5th term + 4d  = 18 + 4(d)

Wait, got it its 1/25

Your sequence is:

(12, 15, 18, 21, 51, 81, 102, 105, 108, 120, 123, 126, 129, 132, 135, 138, 144, 147, 150, 153, 156, 159, 162, 165, 168, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 210, 213, 216, 219, 231, 261, 291, 312, 315, 318, 321, 351, 381, 414, 417, 441, 471, 501, 510, 513, 516, 519, 531, 561, 591, 612, 615, 618, 621, 651, 681, 714, 717, 741, 771, 801, 810, 813, 816, 819, 831, 861, 891, 912, 915)>>Total =80 terms

The 80th term = 915

Uh.. oke 24. 

But nobody came...

You're Welcome!

First, lets find how many ways to get to (4, 3) then from (4, 3) to (5, 7) then subtract that from the ways to go from (0, 0) to (5, 7).

The formula for going from one point two another on a perfect grid is ((x0, y0) is the first point and (x1, y1) is the second point) (abs(x1 - x0) + abs(x1 - x0)) choose abs(x1 - x0) or abs(y1 - y0)

This should be enough to get you started on how to solve this problem.

(Btw, i got it right on aops and the answer is 617) (yes i know this is aops.)