Consider the table:
1 = 1
2+3+4 = 1 + 8
5+6+7+8+9 = 8 + 27
10+11+12+13+14+15+16 = 27 + 64
Guess the general law suggested by these examples. Prove it if you can.
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Posted 04 July 2013 - 04:38 AM Best Answer
Posted 05 July 2013 - 05:02 AM
There are x2-(x-1)2 terms on the left hand side (in general, if you count the whole numbers from n to m, then that's m-n+1 numbers), so we can change the left hand side to:
[First term plus last term] times [number of terms in the sequence] divided by 2
[(x-1)2+1 + x2] [x2-(x-1)2] / 2
Multiplying all that out gives
[(x2-2x+1) + 1 + x2] [x2-(x2-2x+1)] / 2
[2x2-2x+2] [2x-1] / 2
And that can be rewritten as
x3 + (x3-3x2+3x-1) = x3 + (x-1)3
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