MTH202 Assignment No. 3 Spring 2012 solution

Question no1

 Find a common fraction for the recurring decimal .

Answer;

=1.53 + 0.0053 + 0.000053 + 0.00000053 +…

 

The common fraction for 1.53 is 153/99

 

Question 2

Prove by mathematical induction that  is divisible by 2 for all positive integral values of n.

 

Answer;

 

1. Basis Step:

For n = 1

n3 – n = 13 – 1 = 1 – 1 = 0

2.Inductive Step:

Suppose that the statement is true for n = k, i.e., k3-k is divisible by 2

for all n ∈Z+

Then

k3-k = 2·q…………………….(1)

We need to prove that (k+1)3 – (k+1) is divisible by 2.

Now

(k+1) 3 – (k+1) = (k3 + 3k2 + 3k + 1) – (k + 1)

= (k3 – k) + 3k2 + 3k

= 2·q + 3·(k2 + k) using(1)

 

Question 3                                                                                                           Marks: 5

Use the Euclidean algorithm to calculate the greater common divisor of the following pair of integers 504 and 396.

Answers;

504=396.1+108

396=108.3+72

108=72.1+36

72=36.2+0

Now we fined gcd id 36

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