+0

# Find the value of the sum

0
57
4
+467

\(\binom{99}{0} - \binom{99}{2} + \binom{99}{4} - \dots - \binom{99}{98}.\)

Dec 1, 2020

#1
0

By Mathematica, Sum(binom(99,2n)*(-1)^n, 0 <= n <= 49) = 2^50.

Dec 1, 2020
#2
+467
0

incorrect

Dec 1, 2020
#3
+1

sumfor(n, 1, 98, (1 - ((99 nCr 2*n)+(99 nCr 4*n) - (99 nCr 6*n))) ==-56294 9953421312== - 2^49

Dec 1, 2020
#4
+114592
+1

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

Dec 1, 2020