Mathematics

Problem

Let

S_{n}

denote the sum of all the numbers in the

n

th row of the triangular array of binomial coefficients, where

n\ge 0

. a. (i) Show that

S_{n}=2\,S_{n-1}

for every

n\ge 1

. b. (ii) Given that

S_{0}=1

, obtain a closed–form expression for

S_{n}

in terms of

n

. c. (iii) Verify your formula for

n=6

by adding the entries of the 6th row directly.