[RESOLVED] Pattern

[Yunie]

(b)
From (1) in Solution 1, we conclude that the equality r = s + 1 is true. So, we substitute r with s + 1 in the given equality.

(s+1)^3 - s^3 = 331 ..................... (3)

Now, we can use the theorem for the difference of cubes, which states that for any two real numbers x and y, the equality x^3 - y^3 = (x-y) * (x^2+x*y+y^2) is true.

(s+1)^3 - s^3 =
= ((s+1) - s) * ((s+1)^2 + (s+1)*s + s^2) =
= 1 * (s^2 + 2*s + 1 + s^2 + s + s^2) =
= 3 * s^2 + 3*s + 1 ....................... (4)

From (3), (4)
3 * s^2 + 3*s + 1 = 331 <=>
3 * s^2 + 3*s = 330 <=>
3 * (s^2 + s) = 330 / :3 (Divide both sides by 3.) <=>
s^2 + s = 110 <=>
s * (s+1) = 110

At this point, the solution is obvious, s = 10, s + 1 = r = 11.

Thanks again for replying to my threads again. I really do appreciate your help. And, sorry for using the color pink, I will change it now. Sorry for making it difficult for you to read, my deepest apologies.

Hmm, I don't really understand about the bolded parts...Where does r = s +1 comes from? I am really confused about the whole thing in part (b)...Maybe you could somehow explain it in another way? Sorry for the trouble I have caused you, my deepest apologies again. My maths aren't that good so hope you will understand..My exams are coming so hope that I can understand and do well...Thank you again.