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naxos
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#1
Report Thread starter 14 years ago
#1
I have been trying to read through the chapter, but I can't :confused: understand it because the book just skips a load of steps. So could you please help me understand this example:

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.
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Jonny W
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Report 14 years ago
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24r^2 + 2 = (2r + 1)^3 - (2r - 1)^3

(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).
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RichE
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Report 14 years ago
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To get the blue box equation they've jsut added the previous n equations.

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.
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