Proofs, Diophantine equations, and sequences
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1- Prove by induction, the following result for all n
1- Prove by induction, the following result for all n
2- Prove that , if x2
3- Use induction to prove that
(This can be used to show that the infinite harmonic series diverges.)
4- A sequence x1, x2, x3, ......of real numbers is defined by x1=1 and
xn+1 = , if n
Prove that xn = for all positive integers n.
5- A sequence of integers x1, x2, x3, ......is defined by x1=3, x2=7, and
xk = for k
Prove that xn = 2n + 3n-1 for all n .
6- If a = and b = , prove that fn = for all n
7- Find all the integer solutions to the following Diophantine equation:
169x-65y = 91
8- Find all the nonnegative interger solutions to the following Diophantine equation:
12x + 57y = 423
9- Show that the Diophantine equation ax2 + by2 = c does not have any integer solutions unless gcd (a, b)|c. If gcd (a, b)|c, does the equation always have an integer solution?
10- A trucking company has to move 844 refrigerators. It has two types of trucks it can use; one carries 28 refrigerators and the other 34 refrigerators. If it only sends out full trucks and all the trucks return empty. List the possible ways of moving all the refrigerators.
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Solution Summary
This provides examples of proving a variety of statements, using induction, working with Diophantine equations, and working with sequences.
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Hi,
Please find the solution/explanation attached
1- Prove by induction, the following result for all n
Let n = 1, we will get
Left hand side =
Right hand side =
Thus it is true for n = 1.
Now let it is true for n = k
That is
Now we will prove it for n = k+1
That is we will show that
Now,
Now from (i),
Thus, it is true for n = k + 1.
Hence, by mathematical induction it is true for all n.
2- Prove that , if x2
Let n = 1, we will get
Left hand side =
Right hand side =
Thus it is true for n = 1.
Now let it is true for n = k
----------(i)
Now we will prove it for n = k+1
That is we will show that
Now,
Now from (i)
Thus, it is true for n = k + 1.
Hence, by mathematical induction it is true for all n.
3- Use induction to prove that
(This can be used to show that the infinite harmonic series diverges.)
Solution:
Let n = 1, we will get
Left hand side = 1 + ½ = 3/2
Right hand side = 1 + ½ = 3/2
Let it is true for some n = k,
---(i)
We will prove it for n = k + 1
That is
Now,
Thus it is true for n = k + 1.
Hence, by mathematical induction it is true , for all n.
4- A sequence x1, x2, x3, ......of real numbers is ...
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