# Using Mathematical Induction to Prove 2^(3n) – 3^n is divisible by 5

Hello and welcome! In this video, we can use mathematical induction to prove that 2^(3n) – 3^n is divisible by 5 for all natural numbers, or n is greater than

or equal to 1 the first step in mathematical induction is

always the basis test which is to see if this condition or expression holds true for n=1 We put a question mark because we are just going to test if it is true or not, so… if n=1… then I get 2^(3*1) – 3^1… that’s equal to… 2^3=8 and 3^1=3 so I get 8 – 3 which is equal to 5. And

is 5 divisible by 5? Yes it is, so the basis test validates this condition. Now because we

have found that this statement is true for n=1, I’m going to assume that it holds true when n is equal to any integer so I’m going to assume true for n=k so I get 2^(3k) – 3^k=5J So this simply means that on the the

right hand side we have 5 multiplied by any integer or the right hand side basically means I get any multiple five, which is divisible by 5. So I say: J is any natural number. I know that this equation

can imply that 2^(3k)=5J + 3^k and this step will become important later on Ok now, the second major step of

mathematical induction is the inductive test. So I’ve assumed that if this statement is true

for n=k… does is hold true for when n=k + 1? or when I go to the next number? So if I substitute in n=k + 1… I’ll get 2^[3(k+1)] – 3^(k+1) and that can be written as 2^(3k+3) So I’ve just expand… distribute this out…

and I get 3^(k+1) and if I just use my index laws to simplify, I will get… (2^3k)(2^3) – (3k)(3) which is equal to 8(2^3k) – 3(3^k) alright, here’s where… this expression becomes important we have 8[5J + 3^k] – 3(3^k) If I expand that out, I will get 40J + 8(3^k) – 3(3^k) an well 8(3^k) – 3(3^k) that will give me 5(3^k)… and 40J remains… now if I factor out a 5, … I will get inside of brackets… 8J + 3^k well this term inside of the parentheses, or the brackets will always work out to be a whole number if J and k are natural numbers which they are and we have a whole number multiplied by 5 which means it is going to be divisible five. So we have proven by mathematical induction that the expression… 2^(3n) – 3^n is divisible by 5 for all natural numbers. So for all n is greater

than 1 and that’s all we’re required to do. So please give me a thumbs up if this video has helped you to better understand how to use mathematical induction. If you are math student please feel free to subscribe for future

videos that may help you on assignments or exams. If you have any

questions that you would like me to do a video on please feel free to comment on any of the videos seem thanks for watching and I hope you’ve learned something

thanks man that helped a lote

Thank you. Keep up the good work:D

thank you thank you thank you!

Are you doing that with a mouse!? Or is that just ur digitizer hitting the screen?

Seriously man, ur video was very helpful

Could you show a proof using mathematical induction of n^3 – n is divisible by 3 for all n> >=2?

2 to parallel of 3k what??? aaah 2 to power of 3. Sorry your sound confused me….AM i the only one???

I've got my FP1 exam tomorrow and this was really helpful. Thanks man

Thank yo so much for this video Sir !

(10^2n-1+ 1 is divisible by 11) please can u do a video on this problem as well ?

only your video can make me understand this 😂 . thank you soooo much!!

nice dude

prove that if x and y are natural numbers such that k=(x^2 +y^2)/xy is an integer, then k=2

Samwell Tarly.. is that you ?

Nice bro.

n(n+1)(n+5) is a multiple of 6

can you please help me out with this?

btw awesome video 👐 I finally understood this 😂

7^2n-3(5)^n+2 is divisible by 12. Pls help me with these 🙁

tytyty

Was i the only one who forgot what was done at 8:40 ?

Thanks a lot man

THANKS SO MUCH!

nice one..

This helped greatly! You should do more videos since you explain complex proofs so well.

Mathematical induction https://youtu.be/bYxpMwLrdjs

Example 1 https://youtu.be/rbr_MQokZW4

Example 2 https://youtu.be/sB41hF5J138

Example 3 https://youtu.be/75qLooePjNc

Thank you Sir. Now I understand 👏✌✊👍

how to take a common no in this step k divided by 2 [5k+1]+5[k-2]-2,in order to proceed the sum.Please help

Thank you! This helped me so much

thank you very much

Thank you!! I used this strategy to prove 5^(2n) – 2^(5n) is divisible by 7 for all natural n.

bakvas hai , Sahi se do

Thank you very much. I really didn't know that.

Thanks!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

You are The best!!!

thank you

Well put!

or Eka digia

GREAT explanation. Thank you.

Thank you! Your explanation is very clear. I didn't get proof by induction before watching your video!!!! It helped me a lot.

Very helpful. Finally solved my maths paper’s final problem: Prove 4^n +15n – 1 is divisible by 9

literally saved my life

HOW CAN U PROVE THIS ONE 3^N+7^N-2 IS DIVISIBLE BY 8 ;N>=1

Thank you, it is a good example

Thank you Sir, this is the most clearly explained P.M.I ever!!!!!! I wish I could get the same on permutations

i love this vid thank god for you

Absolutely useful. Thanks a lot, I was having a hard time finding an example of how to think the induction when n is used as an exponent

thx

i downvoted because no indian accent.

jk great video !