some induction questions

[vixity]

1)
a) by considering the sum of the terms of an arithmetic series, show that

(1+2+...+n)^2 = n^2/4(n+1)^2

b) By using the principle of mathematical induction prove that

1^3+2^3+...+n^3 = (1+2+...+n)^2, for all n (greater or equal) to 1.

2)
Use mathematical induction to prove that, for every positive integer n,

13*6^n + 2 is divisible by 5.

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can someone show me how do you do it since the term is being squared i can't find the common difference and in 2 i can't make an equation.

[Glaysher]

1)

1 + 2 + ... + n is arithmetic series with a = 1 and L = n

Sum of arithmetic series = (n/2)(a + L)

1 + 2 + ... + n = (n/2)(1 + n)

(1 + 2 + ... + n)² = [(n/2)(1 + n)]²

(1 + 2 + ... + n)² = (n/2)²(1 + n)]²

(1 + 2 + ... + n)² = (n²/2²)(n + 1)²

[Glaysher]

b) n = 1 case true

1^3 = 1 = 1^2

Assume true for n = k case

1^3+2^3+...+k^3 = (1+2+...+k)^2

Try n = k + 1 case

1^3+2^3+...+k^3 + (k + 1)^3 = (1+2+...+k)^2 + (k + 1)^3

= (k^2/4) (k+1)^2 + (k + 1)^3 from part (a)

= (k + 1)^2 [(k^2/4 + k + 1]

= [(k + 1)^2/4](k^2 + 4k + 4)

= [(k + 1)^2/4](k + 2)^2

= (1 + 2 + ... + k + (k + 1))^2 from part (a)

So true for n = k + 1

By principle of mathematical induction true for all integers n greater than or equal to 1

[Glaysher]

2) n = 1 case

13 * 6[sup1[/sup] + 2 = 78 + 2 = 80 = 16 * 5

True for n = 1

Assume true for n = k case

13 * 6^k + 2 = 5m for some integer m

n = k + 1 case

13 * 6^{k + 1} + 2 = 13 * 6 * 6^k + 2

= 78 * 6^k + 2

= 13 * 6^k + 2 + 65 * 6^k

= 5m + 5(13 * * 6^k)

Therefore divisible by 5

True for n = k + 1 case

By principle of mathematical induction true for all integers n greater than or equal to 1

[Dross]

(see attachment)

[DomoCobra]

Ahh good old Proof by Induction

FP1

[vixity]

sheet man you guys are really smart

eurrr fp1?

[vixity]

btw thanks glaysher and dross you made it more easier to understand =D

[Glaysher]

sheet man you guys are really smart

eurrr fp1?

English Further Pure 1 module for A level. However proof by induction is not in all the different exam board's FP1s, EdExcel for example has it in FP3

[Glaysher]

btw thanks glaysher and dross you made it more easier to understand =D

No problem

[vixity]

is ef1 like for the advance people? cause here in australia it is considered as a topic of the advance people(we call it 3 unit) but theres 4 unit where the topics include you know the square root minus 1? imaginary roots or something where you find that out even though you can't have negative square root..yeah topics like that

[Glaysher]

Sounds similar. Only the best students would do further maths modules which contains topics like:

complex numbers, proof by induction, differential equations, hyperbolics and many more

[vixity]

did you do all those further modules????oh wait you probably aced them..

[Glaysher]

The modules as they exist now didn't exist when I did them. I didn't do hyperbolics, and the work on ellipses, parabolas and hyperbolas but I id some group theory instead. All the pure was assessed by a 3 hour exam. I got an A but I did in all my A levels. It was history that gave me the most trouble.

[Dross]

The modules as they exist now didn't exist when I did them. I didn't do hyperbolics, and the work on ellipses, parabolas and hyperbolas but I id some group theory instead.

I'm the other way around - I'm pretty sure the modules taken now (if it's still the AS/A2 line of qualifications) are the ones I took. As it happens, I was part of the guinea pig year, the first year they set them! We didn't do any group theory, but I do remember there were matrices, intrinsic coordinates (which I've never seen used since) and I think polar coordinates were part of the further maths sylabus... not sure though.

Does anybody still know if the AS/A2 syllabus has been kept? Or are my A-Levels all fading away into the trivia section of a quiz somewhere

[Glaysher]

If you took P1 - P6 then no

P1 - P3 have been replaced with C1 - C4

and P4 - P6 have been replaced with FP1 to FP3

[vixity]

wow...you guys sound really old..no offense but....i didn't mean to offend!!!!

[Glaysher]

lol But I've kept my youthful looks so I keep being mistaken for students