Induction Q's (1 Viewer)

[Watsername]

Hey, I have been practacing some induction and I have been getting confused with the last couple lines in some of the questions.

1. Use the method of mathematical induction to show that 9^(n+2) + 4^n is divisble be 5 for all positive integers n

I get down to the line RTP 9^(k+3) - 4^k = 5m then
LHS= 4(9^(k+2) - 4^k) + 5(9^(k+2))
= 4(5m) + 5(9^(k+2)
but then in the solutions it says the next is:
5(4m+9^(k+2)

can someone explain how u get that line? ifu need the whole solution tell me

Thanks

 

[香港!]

Hey, I have been practacing some induction and I have been getting confused with the last couple lines in some of the questions.

1. Use the method of mathematical induction to show that 9^(n+2) + 4^n is divisble be 5 for all positive integers n

I get down to the line RTP 9^(k+3) - 4^k = 5m then
LHS= 4(9^(k+2) - 4^k) + 5(9^(k+2))
= 4(5m) + 5(9^(k+2)
but then in the solutions it says the next is:
5(4m+9^(k+2)

can someone explain how u get that line? ifu need the whole solution tell me

Thanks

HiHi

I'll do it from assume true for n=k...
So assuming the statement is true for n=k, we have:
9^(k+2) + 4^k=5m where m is positive integer
4^k=5m-9^(k+2)--------->(1)
Now to prove it's true for n=k+1;
9^(k+3)+4^(k+1)
=9^(k+3)+4*4^(k)
=9^(k+3)+4(5m-9^(k+2) ) using (1)
=9^(k+3)+20m-4*9^(k+2)
=9^(k+2)(9-4)+20m
=5[9^(k+2)+4m]
which is divisible by 5....

hope that makes it clear

 

[haboozin]

= 4(5m) + 5(9^(k+2)
but then in the solutions it says the next is:
5(4m+9^(k+2)

can someone explain how u get that line? ifu need the whole solution tell me

Thanks

your stuck between those 2 lines? :|

look it says 4(<b>5</b>m) + <b>5</b>(9^(k+2)

common factor of 5... dont be scared if the 5 is in the bracket....

 

[Stefano]

= 4(5m) + 5(9^(k+2)
but then in the solutions it says the next is:
5(4m+9^(k+2)

ROFL !! HAHA

Sorry. I could't help it!

Don't worry, it happens to all of us. It's just funny how mathematics can confuse the mind. Well done though, you got it out.

 

[haboozin]

Don't worry, it happens to all of us. It's just funny how mathematics can confuse the mind. Well done though, you got it out.

he was just having a look at the solutions, and didnt understand how they skipped to that line.