Dear fellows kindly follow the blog by pressing the follow button

Maximum Marks: 12

Due Date: June 02, 2009

DON’T MISS THESE: Important instructions before attempting the solution of this assignment:

•To solve this assignment, you should have good command over 23 -30 lectures.

• Try to get the concepts, consolidate your concepts and ideas from these questions which you learn in the 23 to 30 lectures.

• Upload assignments properly through LMS, No Assignment will be accepted

through email.

• Write your ID on the top of your solution file.

• Don’t use colorful back grounds in your solution files.

• Use Math Type or Equation Editor etc for mathematical symbols.

• In order to get full marks do all necessary steps.

• You should remember that if we found the solution files of some students are same then we will reward zero marks to all those students.

• Try to make solution by yourself and protect your work from other students, otherwise you and the student who send same solution file as you will be given zero marks.

• Also remember that you are supposed to submit your assignment in MS-Word ,any other like scan images etc will not be accepted and we will give zero marks correspond to these assignments.

Question 1; Mark: 7

Use mathematical induction to prove that for all integers n≥1,

is divisible by 3.

5^n-2^n

is divisible by 3.

P(1) is true

For n = 1, L.H.S

of P(5^1-2^1)

= 5-2

= 3

Hence the equation is true for 5^n-2^n is divisible by 3

Base Case: Let n = 0. Then

15^n - 1 = 15^0 - 1 = 1 - 1 = 0, and 0 is obviously divisible by 7.

Let n = 1. Then

15^1 - 1 = 15 - 1 = 14, and 14 is obviously divisible by 7.

Therefore, the formula holds true for n = 0 and n = 1.

Induction Hypothesis: Assume the formula holds true for up to n = k for some value k. i.e. assume that

15^k - 1 is divisible by 7.

(We want to prove that the formula holds true for n = k + 1, i.e. that 15^(k + 1) - 1 is divisible by 7)

BUT, what exactly is 15^(k + 1) - 1 ? Let's evaluate that.

15^(k + 1) - 1 = 15*(15^k) - 1

What I'm going to do is split 15*(15^k) as 14(15^k) + 1(15^k).

= 14(15^k) + 1(15^k) - 1

= 14(15^k) + (15^k) - 1

I'm going to place the last two terms in square brackets to prove a point.

= 14(15^k) + [ (15^k) - 1 ]

Look at this expression; the second half of this expression is divisible by 7 because 15^k - 1 being divisible by 7 is our induction hypothesis.

The first half, 14(15^k), is OBVIOUSLY divisible by 7, since

14(15^k) = (7*2)(15^k).

That means we have the sum of two things divisible by 7, which means as a whole,

14(15^k) + [ (15^k) - 1 ]

is divisible by 7, and the above expression is equal to

15^(k + 1) - 1

Therefore, the formula holds true for n = k + 1.

And thus, by the Principle of Mathematical Induction,

15^n - 1 is divisible by 7 for all integers n >= 0.

Question 2; Mark: 5

A club consists of four memebers.How many sample points are in the sample space when three officers; president, secretary and treasurer, are to be chosen?

Kindly wait for the second questions solution

## No comments:

Post a Comment