Question

Please explain this: 1^2 + 4^2 + 7^2 + ... + (3n - 2)^2 = n(6n^2-3n-1)/2

Answer 1

do you know these ?
\(\sum n = \dfrac{n(n+1)}{2} \\ \sum n^2 = \dfrac{n(n+1)(2n+1)}{6}\)
?

Answer 2

Yes I did that and n= 1

Answer 3

\(\sum c=nc\)

Answer 4

n= 1 ?

Answer 5

To answer the prob it first said I had to prove it equals n=1

Answer 6

oh, the mathematical induction way!?

Answer 7

Yeah , sorry

Answer 8

i was doing it without induction....but oh well...

so, for n=1, you got that true, right ?

Answer 9

Yes

Answer 10

so we assume it true for n= k
1^2 + 4^2 + 7^2 + ... + (3k - 2)^2 = k(6k^2-3k-1)/2

Answer 11

and prove it for n=k+1

Answer 12

so we need to prove
1^2 + 4^2 + 7^2 + ... + (3k - 2)^2+ (3k+1)^2 = (k+1)(6[k+1]^2-3(k+1)-1)/2

Answer 13

Wat I don't get where u get (3k+1)^2 and (k+1)

Answer 14

we need to prove it for n = k+1 right ?
so, 3 (k+1) -2 = 3k+3-2=3k +1
ok?

Answer 15

Ok then y are there two of them Plus and minus

Answer 16

sorry, what exactly is your doubt ?

Answer 17

Y is there both 3k+1 and 3k-1 . I can see how we got the plus not minus

Answer 18

i don't see a 3k-1 here
1^2 + 4^2 + 7^2 + ... + (3k - 2)^2+ (3k+1)^2 = (k+1)(6[k+1]^2-3(k+1)-1)/2

Answer 19

(3k-1)^2

Answer 20

its (3k-2)^2 !
and it will come because we are considering k+1 terms instead of k terms
so
(3k-2)^2 for k'th term
and (3k+1)^2 for 'k+1'th term !
got this ?

Answer 21

I don't understand

Answer 22

i just replaced every 'k' by 'k+1'
thats it!

Answer 23

you replace k by k+1 in
1^2 + 4^2 + 7^2 + ... + (3k - 2)^2 = k(6k^2-3k-1)/2
what u get ?

Answer 24

i still dont get how you get (3k-1)^2

Answer 25

but but i didn't get (3k-1)^2 at all
i had (3k-2)^2 term though...

Answer 26

sorry i think im just overthinking but if you put (3(k+1)-2)^2 it equals (3k+1)^2, so how do you get (3k-2)^2

Answer 27

it will come because we are considering k+1 terms instead of k terms
so
(3k-2)^2 for k'th term
and (3k+1)^2 for 'k+1'th term !
got this ?

Answer 28

thanks i think i get it

Answer 29

sorry if i confused you...

Answer 30

nah im good