Have you been avoiding Proof By Induction until now?
Don't worry, when it comes to the Algebra Proofs, we have you covered.
This blog post highlights some tips and advice on how to approach revision of Proof by Induction.
One important thing to note: It's due to come up (see tip #2), so ignore it at your peril!
Proof by Induction on Tacit Maths
The Tacit Maths module on Proof by Induction gives you:
A notes worksheet with examples of Proof By Induction and Proof by Contradiction.
9 Exam Questions with all the times Proof by Contradiction appeared in LCHL since 2013.
Powerpoint Solutions to ALL of the above.
Marking schemes for all of the Exam Questions, to show you where the marks are going.
Visit the module by clicking the following link:
What are Algebra Proofs?
Have a look at the proof in the image above.
All the values for n shown give you an answer that is divisible by 3.
But how many examples do you need to conclusively prove it is true?
In Maths, we need to be able to prove it's true for EVERY example.
That's where the Algebra Proofs come in handy.
On the Leaving Cert course, the most popular Algebra Proof is Proof by Induction.
There are a few standard Proof by Induction questions (see LC 2014 or LC 2020).
Then you have the "divisible" proofs, which follow the same process.
You can then have Proof by Induction linked to other topics on your course, like De Moivre's Theorem (shown above), or the Geometric Series formula (see LC 2012).
Finally, you have Proof by Contradiction, but as you'll see in a later post, that hasn't appeared in a long time...!
The Bad News #1: Proof by Induction is due to come up in 2023
Proof by Contradiction appeared in the following years:
2012: Prove Geometric Series Formula using Proof by Induction 2014: Proof by Induction (Natural Numbers) 2015: n/a 2016: Proof by Induction ("Divisible") 2017: n/a 2018: Prove De Moivre's Theorem using P. by I.) 2019: Proof by Induction with inequalities 2020: Proof by Induction (Square Numbers) 2021: Proof by Induction ("Divisible") 2022: n/a
The Bad News #2: It's worth a LOT of marks...
As you can see in the image, Proof by Induction is always worth 15 Marks.
That is almost 7% of your entire Paper 1!
If you plan on leaving it out, maybe think again!
On the plus side, you might notice from the marking scheme that it is actually not that difficult to get 8/15 marks or even 12/15.
You just need to know the steps inside out.
More on the steps in the next section...
The Good News: 80% of Proof by Induction is quite routine...
Proof by Induction is made up of 3 steps (as mentioned in the marking scheme) and the Conclusion.
Step 1: Show for n = 1.
Step 2: Assume true for n = k.
These two steps are quite simple.
Step 3: Prove true for n= k +1.
Step 3 is where the magic happens.
But the good news is once you have the first two steps done, you have already picked up most of your marks.
And don't forget your conclusion!
Finally...a prediction: Proof by Contradiction
Proof by Contradiction follows a similar format to Proof by Induction, but the steps are quite different.
It is definitely not something that you could make up on the spot.
It is well worth preparing for however, as it hasn't appeared in the LCHL since 2012!
For more Tips and Advice for teachers and students on how to tackle different topics in the Leaving Cert Higher Level Course, visit our blog.