Question

What is the proof for the sum of all rows above the nth row in the Pascal triangle is one less than the nth row?

Answer 1

here is the triangle
1 ---- sum is 2^0 = 1
1 1 ---- sum is 2^1 = 2
1 2 1 ---- sum is 2^2 = 4
1 3 3 1 ---- sum is 2^3 = 8
1 4 6 4 1 ---- sum is 2^4 = 16

notice that 1+2+4+8 differs from 16 by 1

notice that this sum forms a GS with a=1, r =2
so that the sum(n) = 1(2^n - 1)/(2-1) = 2^n - 1
term(n+1) = 2^n
= 1(2^n - 1) = 2^n - 1
they differ by 2^n - (2^n - 1)
= 2^n - 2^n + 1
= 1