Let k-1 be the first coefficient we are looking for in the nth row of Pascal's Triangle
The second coefficient will be the kth element on row n
And the last coefficient will be the (k + 1) st element on row n
We have seen that C(n , k) / C( n, k -1) = (n - k + 1) / k
So...we can calculate C( n, k + 1) / C ( n ,k) as
n! ( n - k)! k!
________________ * _________ =
[ n - (k+ 1)]! (k+ 1)! n!
( n - k)! k!
__________ * ______ =
[ n - (k + 1)]! (k + 1)!
( n - k)! 1
________ * _____ =
(n - k - 1)! k + 1
n - k
________
k + 1
So we have these two equations
(n - k + 1) / k = 7
(n-k) / (k + 1) = 5 simplify
n - k + 1 = 7k
n - k = 5 (k + 1)
n - k + 1 = 7k
n - k = 5k + 5
n - 8k = -1
n - 6k = 5 subtract these
-8k + 6k = - 6
-2k = - 6
k = 3
And
n - 8k = -1
n -8(3) = -1
n - 24 = -1
n = 23
Check :
The coefficients should be
C (n, k -1), C( n , k) and C( n, k + 1) =
C (23 , 2) , C (23, 3) and C (23, 4) =
253 , 1771 and 8855
The ratios are
1 : 1771/253 : 8855/253 =
1 : 7 : 35
