On this page, you should learn to
Make simple algebraic deductive proofs Use the symbol \(\equiv\) for identities Here is a quiz that practises the skills from this page
Complete the following to "prove that 2 consecutive integers add together to make an odd integer"
n 2n + 1 2n n + 1
\(n + (n + 1) \equiv2n+1\)
2n 2n + 1
Work out a b
We can complete the square:
x² - 4x + 7 \(\equiv\) (x - a b
A square number is always greater than or equal to zero, so (x - a
Since, b
(x - a b
Hence x² - 4x + 7 > 0
This is a good technique for showing that a function is not negative
Complete the following to "prove that adding 2 consecutive odd numbers gives a multiple of 4"
2n 4n 2n + 1 4n - 1
4n
Complete the following to "prove that if m and n are either both even or both odd, then m + n is even"
k 2k - 1 j + k j + k - 1 2k
Complete the following to "prove that one more than the product of two consecutive odd integers is the square of an even number"
2n² 2n 2n + 1 4n² 2n - 1
Finding the product means that we multiple the two numbers together
a) Verify that \(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2=4x^3 \) for x = 3
b) Prove that \(x^2\left(x+1\right)^2-\left(x-1\right)^2x^2\equiv4x^3 \) for all x
b) Start with the left hand side. Expand brackets and simplify
a) Verify that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}=\frac{2x+1}{x^2\left(x+1\right)^2} \) for x = -2
b) Prove that \(\frac{1}{x^2}-\frac{1}{\left(x+1\right)^2}\equiv\frac{2x+1}{x^2\left(x+1\right)^2} \) for all x
b) Start with the left hand side, add the algebraic fractions together by making the denominators the same.
Prove that the sum of three consecutive integers is divisible by 3
Let the first integer be n
What would be the value of the next integer?
a) Verify that x² - 4x + 5 is positive for x = -1
b) Prove that x² - 4x + 5 is positive for all x
A square number cannot be negative. Is it posible to write this expression as a square?
Prove that the difference between the square of any two consecutive odd integers is divisible by 8
An odd number can be written as 2n - 1.
A number is divisible by 8 if it has a factor of 8
a) Verify that \(^3 C_1\ +\ ^3 C_2\ =\ ^4C_2\)
b) Prove that \(^{n-1} C_{r-1}\ +\ ^{n-1} C_r\ =\ ^nC_r\)
\(^nC_r=\frac{n!}{(n-r)!r!}\)
Some careful manipulation is required for this proof. Consider that \(r!=r\cdot(r-1)!\)
MY PROGRESS
Self-assessment How much of Deductive Proofs have you understood?
My notes
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On this page, we will look at deductive reasoning in order to be able to make direct proofs. This is a hugely important topic in mathematics, since we like to be absolutely sure of the results we have found. However, this can be a challenging - when a problem is unfamiliar, it is often difficult to know where to start. The key is to be able to use our existing knowledge and theorems, make connections and work towards a conclusion. You might want to think of these as puzzles to solve.
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