Induction Inequality Proof Example 3: 5^n + 9 less than 6^n

Another viewer-submitted question. Inequality proofs seem particularly difficult when they involve powers of n, but they can be managed just like any other inequality given the right algebraic techniques!

I had to stop and start this video a couple of times due to the cleaner walking into my staffroom while I was shooting. Hopefully it doesn't interrupt the flow of the explanation too much!

[sdadasdadsada]

Sir you deserve a spot in HEAVEN for this one, literally didn’t get this for the longest time until I stumbled upon this jewel

[jmaham]

Awesome explanation, thanks. Made me think about as opposed to blindly following steps that do not always work.

[nickmeyer]

I'm generally solid with logic problems and proofs, but this one stumped the heck out of me. I re-read my material for hours, searched online, and worked with classmates. I came to a solution, but I was never able to fully understand why it worked, until I found this video. Fantastic explanation, and done in a way that clearly explains the concepts without the viewer feeling stupid. Thank you!

[steveochoa]

Amazingly skilled instructor with great demeanor and tone. Easy to pay attention and follow. Wish I could have learned from him in high school.

[mattbenningfield]

Didn't understand mathematical induction with inequalities until I watched this video!  Amazing job on explaining the individual steps to obtain the end result!  Thank you!

[fieryjack]

This is pure wizardry and black magic!:) Brilliant explanation, a real pleasure to follow!

[WillSkForFood]

Just stumbled upon your videos! You are a truly gifted teacher! Thanks so much for sharing this and taking the time to make your videos!

[dahudge]

I'm struggling with my Discrete Maths homework and low and behold this is the exact problem I'm struggling with out of my book. Thanks so much!

[FiftiesDad]

Best description I have seen so far. Thank you very much.

[ernestc.mutalejr]

7 years ago you made a video to save my life today

Again I paused at the beginning and again I used the trick that if the gap between the two side grows continuously, then the equality will always be true. Here's just the inductive step this time:

5^n*5-5^n<6^n*6-6^n
5^4n<6^5n
625^n<7776^n | n-th root (n≠0)
625<7776
I find this MUCH easier than even starting to think about how to approach this with the inequality itself and it works almost every time.

[chuckhoush]

Holy cow THANK YOU this makes a lot more sense now

[avidgamer]

To make the last step easier, you could've just added in a new inequality to show what you were saying is true.

i.e.  5^(k+1) + 9 < 5^(k+1) + 9 + 5^k + 45

Then from that inequality you could easily show that since:
5^(k+1) + 9 + 5^k + 45 < 6^(k+1)

Then 5^(k+1) + 9 < 6^(k+1)

Both results are the same. This is simply better for showing your thought process and work for anyone who has to do this type of induction on a test.

[Yeldho]

This man is a godsend. Hoooly, i understand it way better now. Thank you so much!

[Supercatzs]

I finally understand inequality proofs now, thank you so much

[markxphillies]

my problem was very different but similar enough to the point where i could easily apply your steps and change them to fit my problem. thank you so much!

[cliftonmccallum]

Thank you for the detailed explanation; I finally understand this.

[knickkennedy]

Cleanest explanation I've found on this topic. Thanks!

[ezodragon]

Thank you! Inequality induction is giving me huge troubles and I've finally understood an example. 

[mistbabe]

Your excellent, please continue to make great videos such as this. This video saved me for my finals which is tomorrow :(.
Thank you Eddie!!!