I tried to do these questions, however I can't seem to get the answer. Can someone help me? I would really appreciate it.
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Mathematical Induction
Prove that the statement in true for every positive integer n.
1. 1/1*2*3 + 1/2*3*4 + 1/3*4*5 + .... + 1/n*(n+1)*(n+2)
2. 1+2*2+3*2^2+...+n*2^n-1= 1+(n-1)*2n
3. 9 is a factor of 10^n+1+3*10^n+5
Partial Fraction Decomposition
1. 3x^3+11x^2+16x+5
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x(x+1)^3
2. 2x^4-2x^3+6x^2-5x+1
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x^3-x^2+x-1
Sure, I can help you with these questions! Let's go through each of them step by step and explain how to find the answer.
Mathematical Induction:
To prove that a statement is true for every positive integer n using mathematical induction, we typically follow these steps:
1. Base Case: Start by proving the statement for the smallest possible value of n (usually n = 1 or n = 0).
2. Inductive Hypothesis: Assume that the statement is true for some arbitrary positive integer k.
3. Inductive Step: Using the assumption in the previous step, prove that if the statement is true for k, then it must also be true for k + 1.
4. Conclusion: By completing the base case and inductive step, we can conclude that the statement is true for all positive integers n.
Now, let's apply these steps to the provided questions.
1. To prove 1/1*2*3 + 1/2*3*4 + 1/3*4*5 + .... + 1/n*(n+1)*(n+2) is true for every positive integer n.
- Base case: Substitute n = 1 and calculate the value of the expression to check if it holds true.
1/1*2*3 = 1/6
- Inductive hypothesis: Assume the statement is true for some arbitrary positive integer k.
Suppose, 1/1*2*3 + 1/2*3*4 + ... + 1/k*(k+1)*(k+2) = (k+1)*(k+2)/6
- Inductive step: Now, we need to prove that if the statement is true for k, then it must also be true for k + 1.
Substitute n = k + 1 and simplify the expression as follows:
1/1*2*3 + 1/2*3*4 + ... + 1/k*(k+1)*(k+2) + 1/(k+1)*(k+2)*(k+3)
= [(k+1)*(k+2)/6] + 1/(k+1)*(k+2)*(k+3)
You can simplify the expression on the right-hand side to show that it is equal to (k+2)*(k+3)/6.
This would imply that the statement is true for k + 1.
- Conclusion: By completing the base case and inductive step, we have proven that the statement is true for every positive integer n using mathematical induction.
You can apply the same approach to the other two mathematical induction questions.
Partial Fraction Decomposition:
Partial fraction decomposition is a method to express a rational function as the sum of simpler fractions.
Let's do the first partial fraction decomposition question:
1. To decompose 3x^3 + 11x^2 + 16x + 5 / (x(x+1)^3)
- First, factorize the denominator: x(x+1)^3 = x(x+1)(x+1)(x+1).
- Now, we need to determine the decomposition of the numerator. We'll have to find constants A, B, C, and D such that:
(3x^3 + 11x^2 + 16x + 5) / (x(x+1)^3) = A/x + B/(x+1) + C/(x+1)^2 + D/(x+1)^3
- To find the values of A, B, C, and D, multiply the entire equation by the denominator x(x+1)^3:
3x^3 + 11x^2 + 16x + 5 = A(x+1)^3 + Bx(x+1)^2 + Cx(x+1) + Dx
- Now, you need to equate the coefficients of like terms on both sides of the equation.
Start by expanding (x+1)^3 and multiplying out all the terms.
- After equating the coefficients, you will end up with a system of equations. Solve this system to find the values of A, B, C, and D.
Once you find the values of A, B, C, and D, you can substitute them back into the initial equation to express the given rational function as a sum of simpler fractions.
You can follow a similar approach for the second partial fraction decomposition question.
I hope this explanation helps you to understand how to solve these questions! Feel free to ask if you have any further doubts.