mth202 assignment 2 solution

Assignment # 2 Mth202

Spring 2015

Maximum Marks: 20

Due Date: 24^th July, 2015

INSTRUCTIONS:

Please read the following instructions before attempting the solution of this assignment:

• To solve this assignment, you should have good command over 23-29 lectures.

In order to solve this assignment you have strong concepts about following topics:

ü Mathematical Induction

ü Methods of proof & proof by contradiction

ü Algorithm and division algorithm

ü Combinatorics

Try to get the concepts, consolidate your concepts and ideas from these questions which you learn in these lectures. You should concern the recommended books for clarification of concepts.

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Section 1

Section Incharge Name:- Asmat Batool

Download full solution from here

Question: 1 Marks: 10

Prove the following by using the principle of Mathematical Induction:

, For all integers

Question: 2 Marks: 06

Prove by contradiction that

is irrational.

Question: 3 Marks: 04

Name the four properties which a loop with guard G needs to satisfy in order to be true with respect to its pre and post conditions.

Section 2

Section Incharge Name:- Saima Shafi

Question: 1 Marks: 10

Prove the following by using the principle of Mathematical Induction:

Question: 2 Marks: 05+05

(a) A bank customer can only access his locker if he enters his code consists of from one to six digits (repetition is allowed). Then how many different codes are possible?

(b) Find the number of n ways that a team consisting of 12 members can select a coach, assistant coach, and captain. Assume that no member is selected more than one.

Section 3

Section Incharge Name:- Jamshaid Nasir

Question: 1 Marks: 10

Prove the following by using the principle of Mathematical Induction:

Question: 2 Marks: 10

Use the Euclidean algorithm to find GCD (108, 30)

Prove the following by using the principle of Mathematical Induction:

ANS:-

According to the principles of mathematical induction

Show true for n = 1

Assume true for n = k

Show true for n = k + 1

Conclusion: Statement is true for all n >= 1

First showing that equation is true for n=1

⅓+⅓^2+ …………...⅓^n=½(1-⅓^n)

⅓^1=½(3-⅓)

⅓=½(⅔)

⅓=⅓

l.h.s=r.h.s

thus the equation is true for n=1

now using mathematical induction to prove this equation

when n=k

⅓+⅓^2+...................⅓^k=½(1-⅓^k+1)

we have to prove that

⅓+⅓^2+...............⅓^k+1=½(1-⅓^k+1+1)

as,

⅓+⅓^2+................⅓^k+⅓^k+1=½(1-⅓^k+!)

taking l.h.s of equation

as ⅓+⅓^2+................⅓^k=½(1-⅓^k+1)

so it will be

½(1-1/3^k+1)+⅓^k+1

there is a doubt in first one so plzz dont copy it

GCD(108.30)

divide 108 by 30
this gives 108=30.3+18
divide 30 by 18
this gives 30=18.1+12
divide 18 by 12
this gives 12=6.2+0

Hence GCD (108,30) =6

Posted by Arslan

Tuesday, July 21, 2015