Series - (P4/FP1) - help Watch
In the attachment is example from heinamann p4 book on page 13. The things where I can't follow where they came from are underlined in Red . The Blue box is mainly what I don't understand.
Thanks in advanced.
(sum from r = 1 to n) 24r^2 + 2 = (sum from r = 1 to n) (2r + 1)^3 - (2r - 1)^3 . . . . . (*)
LHS of (*) = 2n + 24 (sum from r = 1 to n) r^2
The "2n" comes from adding "2" n times.
RHS of (*) = (2n + 1)^3 - (2*1 - 1)^3 = (2n + 1)^3 - 1
There are 2n cubes in the sum on the RHS, but (2n - 2) of them cancel in pairs (eg: 3^3 and -3^3, 5^3 and -5^3).
On the LHS you have all the squares added together and the 2n comes from addind n lots of 2.
On the RHS the cubes mainly cancel - you have a -3^3 in one equation cancelling with the 3^3 in the next and so on. The only ones that don't cancel are the -1^3 at the start and the cube at the end.