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prove question, i h8 these lol help pls

prove algebraically that the sum of the squares of any two consecutive even integers is never a multiple of 8....4 marks

Reply 1

Lucien_Roach

10

prove algebraically that the sum of the squares of any two consecutive even integers is never a multiple of 8....4 marks

(2x)^2+(2x+2)^2 = 8x^2+8x+4

8(x^2+x+0.5) therefore not divisible by eight

the +4 makes it multiples of 8.5 sorry my first thing ignored the squares

Reply 2

nota bene

15

a^2+(a+1)^2 = 2a^2+2a+1, which is uneven.

edit: or am I totally missing something?

Reply 3

yusufu

16

Lucien_Roach

prove algebraically that the sum of the squares of any two consecutive even integers is never a multiple of 8....4 marks

(2x)+(2x+2) = 4x+2

i dunno how to put this into algebra but all the multiples of eight are multiples of four so if you add 2 to it it will never be a multiple of eight

:smile:

it will either be 2 higher or two lower

It asks about their squares.

Similar method though:
Since numbers working with are even can write as 2x (x E R)
(2x)^2 + (2x+2)^2 = 4x^2 + 4x^2 + 8x + 4
= 8[x^2 + x + 1/2]

Dividing by 8 gives: x^2 + x + 1/2

where x^2 and x are both integers. Adding the half makes sure it is not an integer. Hence the sum is not an integer multiple of 8.

Reply 4

stordoff

1

(2n)^2 + (2n+2)^2

= 4n^2 + 4n^2 +8n + 4

= 8n^2 + 8n + 4

= 8(n^2 + n) + 4

We have a multiple of 8, and then add 4 to it, so no longer have a multiple of 8

Reply 5

tushar_1

OP

4

i thought it is:

n^2 + (n+2)^2 = 2n^2 + 4n + 4

but i dont know how to say this shows it isnt a multiple of 8

can any1 do this? or is my workings out wrong?

Reply 6

Rabite

8

Nota - The next even number will be a+2, won't it?

Let n=2m so that it's even.

(2m)²+(2m+2)²=8(m²+m)+4 which is four more than a multiple of 8, and therefore not a multiple of 8.

[edit]Oh, beaten. Stordoff did the same thing.

Tushar, just let n=2m since it's even then add a bit of explanation, and your proof is complete enough :p:

Reply 7

tushar_1

OP

4

oh i c, thanks ppl, b4 my last post

Reply 8

Lucien_Roach

10

nota bene

a^2+(a+1)^2 = 2a^2+2a+1, which is uneven.

edit: or am I totally missing something?

two consequencutive (sp?) evens

Reply 9

nota bene

15

Rabite

Nota - The next even number will be a+2, won't it?

Ah, it said even in the question as well :rolleyes:

.

I think this is the second question I've not read properly today:/ I'm hopeless.

Reply 10

tushar_1

OP

4

thanks again, gd luck every1 wit their exams

Reply 11

tushar_1

OP

4

going sleep now, gd nyt, swt dreams about maths lol

Reply 12

generalebriety

16

Lucien_Roach

prove algebraically that the sum of the squares of any two consecutive even integers is never a multiple of 8....4 marks

(2x)^2+(2x+2)^2 = 8x^2+8x+4

8(x^2+x+0.5) therefore not divisible by eight

the +4 makes it multiples of 8.5 sorry my first thing ignored the squares