induction question and parametrics (1 Viewer)

[niall045]

Hey, is anyone able to summarise the concept of parametrics for me. Im fairly lost on the subject n have a whole question of it in my half yearlys. If anyone can help itd be much appreciated.

As well, if anyone wants to do step 2 of induction n help me with this question from the 2001 hsc feel free.

Prove by induction that n^ 3 +(n +1)^3 +(n +2)^3 is divisible by 9 for n =1, 2, 3, …



THANKYOU

 

[Estel]

1. Use a textbook and stop being so lazy.
2. (k+1)^3 + (k+2)^3 + (k+3)^3 = 9m + 9k^2 + 27k + 27 by assumption = 9(m+k^2+3k+3)

 

[furryfish]

n^ 3 +(n +1)^3 +(n +2)^3 is divisible by 9 for n =1, 2, 3

Assuming true for n=k,
ie k^3 + (k+1)^3+(k+2)^3 = 9m, for some integer m
Prove true for n=k+1
ie (k+1)^3 + (k+2)^3+(k+3)^3 is divisible by 9.

(k+1)^3 + (k+2)^3+(k+3)^3 = 9m - k^3 + (k+3)^3
= 9m + 9k^2 + 27k + 27
= 9(m+k^2+3k+3), which is divisible by 9

Therefore since the statement is true for n = k, it is also true for n = k + 1.

exams were a while ago, so hopefully thats right.

 

[niall045]

Cheers for that.