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Maximum Marks: 15
Due Date: 15-01-2013
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Q#01: Use mathematical induction to prove that for every integer with . (Note that this inequality is false for and .) Marks = 6
Q#02: Find the GCD of and using Division Algorithm. Marks = 4
Q#03: Define a sequence by the formula , for all integers . Show that this sequence satisfies the recurrence relation , with and for all integers . Marks = 5
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Please Discuss here about this assignment.Thanks
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All students must share points and also your problems.... so it can discussed and solution can prepare..
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maine question 1 kiya hai ye hai
question 1 ka solution:
BASIC STEP:
p(4) = 2^n
2^4 =16 < 24 = 4!
INDUCTIVE STEP:
For this step we assume that p(k) is true for posiive integer k with k>=4
Assume:
2^k < k!
2^k+1 < (k+1)!
2^k+1 = 2.2^k
< 2.k!
< (k+1)k!
= (k+1)!
So p(k+1) is true when p(k) is true.
HENCE PROVED.....
solution of question 2:
Let a= 4566891 and b=182 . Also, let's introduce the variable "r" for the remainder
Let's evaluate a/b . It turns out that a/b = 25092 remainder 147. What we're are interested in is the remainder.
So let . Now assign the value of "b" to "a" and the value of "r" to "b".
So now a=182 ,b=147 and r=147
Now we must ask ourselves: "Does b=0?". Since b=147 (which is NOT zero), we must start all over and keep going until "b" equals zero.
Let's evaluate a/b . It turns out that a/b= 182/147 = 1 remainder 35. What we're are interested in is the remainder.
So let . Now assign the value of "b" to "a" and the value of "r" to "b".
So now a=147 ,b=35 , and r=35
Now we must ask ourselves: "Does b=0?". Since b=35 (which is NOT zero), we must start all over and keep going until "b" equals to zero
Let's evaluate a/b . It turns out that a/b= 147/35 = 4 remainder 7. What we're are interested in is the remainder.
So let . Now assign the value of "b" to "a" and the value of "r" to "b".
So now a=35 ,b=7 , and r=7
Now we must ask ourselves: "Does b=0?". Since b=7 (which is NOT zero), we must start all over and keep going until "b" equals zero.
Let's evaluate a/b . It turns out that a/b = 35/7 = 5 remainder 0. What we're are interested in is the remainder.
So let . Now assign the value of "b" to "a" and the value of "r" to "b".
So now a=7 ,b=0 , and r=0
Since the value of b is now zero, we can stop the process.
Now the value of "a" in the last row is the GCD of the numbers "a" and "b"
So this means that GCD(4566891,182)=7.
.
good work
thnx
great work Sw8 & Sal8y
thnx
nice job
plzzz give some idea for the question no 3 ........................... and thnku soo much Sw8 & Salty for ur help
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