BMAD Posted July 4, 2013 Report Share Posted July 4, 2013 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. Quote Link to comment Share on other sites More sharing options...
0 Barcallica Posted July 4, 2013 Report Share Posted July 4, 2013 Every row can be put as [x^2-(2x-2)]+[x^2-(2x-1)]+......+(x^2-2)+(x^2-1)+x^2=(x-1)^3+x^3 which converts to {[x^2-(2x-2)]+x^2}/2=(x-1)^3+x^3 thus can be proved after some easy simplification. Quote Link to comment Share on other sites More sharing options...
0 Smash72 Posted July 4, 2013 Report Share Posted July 4, 2013 Each sequence of numbers totals the last and next cube number. 17+18+19+20+21+22+23+24+25 = 64+125 26+27+28+29+30+31+32+33+34+35+36 = 125+216 Quote Link to comment Share on other sites More sharing options...
0 plasmid Posted July 5, 2013 Report Share Posted July 5, 2013 The general form of that pattern is Sum from (x-1)2+1 to x2 = (x-1)3 + x3 Make use of the fact that the sum of a series where each term increases or decreases by a fixed amount from the previous term can be rewritten as (the first term plus the last term) times (the number of terms in the series) divided by 2. If you didn't already know that, then the intuitive way of thinking about it is: (the first term plus the last term) is equal to (the second term plus the next-to-last term) which is equal to (the third term plus the third-from-the-end term) etc, so you can rearrange a sum of N numbers into N/2 pairs like that. 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 [x2-x+1] [2x-1] 2x3-3x2+3x-1 And that can be rewritten as x3 + (x3-3x2+3x-1) = x3 + (x-1)3 Quote Link to comment Share on other sites More sharing options...
Question
BMAD
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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