Maths Induction Question (1 Viewer)

[clintmyster]

5²ⁿ - 1 is a multiple of 24
Use induction to prove divisibility for all positive integers n

 

[richiew]

5<SUP>2n</SUP> - 1 = 24p for n > 0, where p is some integer

i) test for n = 1

LHS = 25 - 1
= 24 which is dividisble by 24 therefore true for n = 1

ii) assume true for n = k

i.e. 5<SUP>2k</SUP> - 1 = 24q where q is some integer

iii) prove for n = k+1

5<SUP>2(k+1)</SUP> - 1 = 24q
LHS = 5<SUP>2(k+1)</SUP> - 1
= 5<SUP>2k+2</SUP> - 1
=5<SUP>2</SUP>x5<SUP>2k</SUP> - 1 ----- (1)

from assumption; 5<SUP>2k</SUP> - 1 = 24q therefore
5<SUP>2k</SUP>= 24q +1 ------- (2)
substitute (2) into (1)

LHS = 5<SUP>2</SUP>(24q +1) - 1
= 25 x 24q + 25 - 1
= 25 x 24q + 24
= 24 (25q + 1)
= 24q since (25q + 1) is some integer
therefore true for n = k+1

iv) since result is true for n = 1, and then from step 3 is true for n = 1+1 = 2, and then for n = 2+1 = 3 and so on for all positive integral of n.

yea step (iv) has never made sense to me but my teacher told me write that hahah. but everything else should be right but check over it just to be sure.

hope that helps

 

[clintmyster]

5²ⁿ - 1 = 24p for n > 0, where p is some integer

i) test for n = 1

LHS = 25 - 1
= 24 which is dividisble by 24 therefore true for n = 1

ii) assume true for n = k

i.e. 5^2k - 1 = 24q where q is some integer

iii) prove for n = k+1

5^2(k+1) - 1 = 24q
LHS = 5^2(k+1) - 1
= 5^2k+2 - 1
=5²x5^2k - 1 ----- (1)

from assumption; 5^2k - 1 = 24q therefore
5^2k= 24q +1 ------- (2)
substitute (2) into (1)

LHS = 5²(24q +1) - 1
= 25 x 24q + 25 - 1
= 25 x 24q + 24
= 24 (25q + 1)
= 24q since (25q + 1) is some integer
therefore true for n = k+1

iv) since result is true for n = 1, and then from step 3 is true for n = 1+1 = 2, and then for n = 2+1 = 3 and so on for all positive integral of n.

yea step (iv) has never made sense to me but my teacher told me write that hahah. but everything else should be right but check over it just to be sure.

hope that helps

omg now i know what i did wrong! When I substituted k+1 I didnt put it in brackets! Thanks heaps mate

 

[richiew]

hah yea its those little things that frustrate me too. but yea anytime =)

 

[n_otesguy]

richiew, step (iv) is something that BOS likes...probably to show that you know what proof by induction is.

But if you go onto uni maths, lecturers don't like you writing essays about what proof by induction is. They would only expect you to write (By induction 5<SUP>2n</SUP> - 1 is true for n>0).

 

[richiew]

but would it matter what we write in the HSC? i know the other class in my school got taught to write something simple like that, but i dont recall having problems just regurgitating that at the end of induction questions in past tests.

 

[n_otesguy]

For the purpose of the HSC, it's just best to write it the long winded way because most teachers/markers go by the textbooks written for the HSC.

 

[Kujah]

Yeah I'm doing the long way, but the marker's notes have always said that we don't need to that step, just write that it must be true for all n>/ w/e. :S

 

[clintmyster]

My teacher tells me to regurgitate this:

As it is true for n=1, it will hold true for n=2,3,4... and all positive integers of n!

 

[duy.le]

although i have called up the board to ask this very question the guy did say, they do not award marks for this but simply take away marks if its missing.

so what u write wont really matter as long as it sorta makes a bit of sense. dont make it too crap cause i might get the marker in a bad mood.

my response

"as it is true for n=1 and it is true for n=k+1 if it is true for n=k, it follows by the principle of mathematical induction that it is true for all n>=1"