Mathematical Induction Algebra simplification? (1 Viewer)

cchan334

Member
Joined
Nov 16, 2020
Messages
29
Gender
Female
HSC
2021
So usually for mathematical induction after you assume true for any integer k, and then you try and prove for k+1, and sub in the RHS, it's some easy algebra manipulation and they end up being equal. I don't know why but I've been staring at this question for 20 minutes and I can't seem to be able to do that. It seems that the only way to prove that it is true is by expanding both sides of the equation???

20201210_192539.jpg
 

xilbur

New Member
Joined
Feb 8, 2020
Messages
9
Gender
Male
HSC
2020
(2(k+1)-1)^3 = (2k +2 - 1)^3
= (2k +1 )^3

Is that where you were stuck at?

If you're going straight from the assumption and doing LHS = RHS, be wary of expanding that k+1!
 

cchan334

Member
Joined
Nov 16, 2020
Messages
29
Gender
Female
HSC
2021
(2(k+1)-1)^3 = (2k +2 - 1)^3
= (2k +1 )^3

Is that where you were stuck at?

If you're going straight from the assumption and doing LHS = RHS, be wary of expanding that k+1!
I expanded it already, but like isn't there an easier way than brute expanding it because usually idk with some algebra manipulation I can show that they are equal without expanding it???
 

cossine

Well-Known Member
Joined
Jul 24, 2020
Messages
583
Gender
Male
HSC
2017
You are almost there.

You wrote (2(k+1) -1)^3 as (2k-1)^3 instead of (2(k+1) -1)^3 as (2k+1)^3 as mentioned by xilbur.

From there it should be an easy direct proof.

Note: Make sure to prove true for n=1 and then assume true for n=k. Leaving these steps out might lead to deduction of marks
 

cchan334

Member
Joined
Nov 16, 2020
Messages
29
Gender
Female
HSC
2021
Oh my bad. I did the proof by brute expansion on both sides but I felt like there had to be an easier way?16075932445711297970395323032375.jpg
Like is the only method for showing the LHS =RHS through expanding both sides cas I really can't see it being done any other way, but then that seems to tedious
 

Trebla

Administrator
Administrator
Joined
Feb 16, 2005
Messages
8,113
Gender
Male
HSC
2006
Expanding and simplifying is the most efficient approach here. It’s not that bad lol.

If you insist on not expanding cubics but happy to expand quadratics then the following manipulation can be done, but it’s longer and trickier than fully expanding.

 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top