Answer to Question #98921 in Discrete Mathematics for Dristanta Wagle

Question #98921
Xn
i=1
i2
i = (n − 1)2n+1 + 2
1
Expert's answer
2019-11-19T09:27:25-0500

Using induction to prove


We need prove a statement "\\sum_{i=0} ^ {n} i \\times 2^i = (n - 1) \\times 2^{n+1} + 2"


Proof:


for n = 1


LHS =

"\\sum_{i=0} ^ {1} i \\times 2^i = 2"




RHS =

"(n - 1) \\times 2^{n+1} + 2 = 0 + 2 = 2"


"LHS = RHS"


The given statement is true for n= 1


Let the given statement is true for n = k (integer)



"\\sum_{i=0} ^ {k} i \\times 2^i = (k - 1) \\times 2^{k+1} + 2"




Now, Let n = k +1



"\\sum_{i=0} ^ {k+1} i \\times 2^i = \\sum_{i=0} ^ {k} i \\times 2^i + ( k+1 ) \\times 2^{k+1}"

"= (k - 1) \\times 2^{k+1} + 2 + (k+1 ) \\times 2^ {k+1}"

"= k \\times 2^ {k+1} - 2^ {k+1} + 2 + k \\times 2^{k+1} + 2^{k+1}"


"= 2k \\times 2^ {k+1} + 2 = k \\times 2 ^ {k+1+1} + 2"

"\\sum_{i=0} ^ {k+1} i \\times 2^i = (k+1 -1) \\times 2^ {k+1+1} + 2"

The given statement is true for n = k+1. By the principle of mathematical induction, it was proved that


"\\sum_{i=0} ^ {n} i \\times 2^i = (n - 1) \\times 2^{n+1} + 2"


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS