+466 \(\binom{99}{0} - \binom{99}{2} + \binom{99}{4} - \dots - \binom{99}{98}.\)
sumfor(n, 1, 98, (1 - ((99 nCr 2*n)+(99 nCr 4*n) - (99 nCr 6*n))) ==-56294 9953421312== - 2^49
+128408 Let's see if we can detect a pattern ....
Row n 1 3 5 7 9 11 13 15 17
Sum of terms in odd position 1 -2 -4 8 16 -32 -64 128 256
Note that the alternating sums of the odd position terms in the odd rows seem to follow the pattern
2^[ (n -1)/2 ] where n is the row (ignoring the signs on the sum)
So....the 99th row should have the sum 2^[ (99 -1)/2] = 2^49 (ignoring the sign)
And note that starting with row 3, the signs on these sums change after every 4th term (row)
So ...on the 99th row....we have the [ (99-3) / 4] = [ 96/4 ] = 16th sign change (after the sign on the 3rd row sum of the terms in the odd positions)
So....the sign on the sum should be (-1)^(number of sign changes - 1) = (-1)^(16 - 1) = (-1)^15 = -1
So....the sum is (-1)(2)^49 = -2^49
