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 Shapes are too Deep (Posted on 2005-06-01)
What are the next three numbers:

1, 2, 4, 7, 13, 23, 39, 64, 99, 150, 216, __, __, __?

 See The Solution Submitted by Dustin Rating: 3.6000 (5 votes)

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 Puzzle Solution Comment 5 of 5 |

`Let the ith term of the given sequence be denoted by T(i).`
`Let us define:D(i) = T(i+1) - T(i), and:D_2(i) = D(i+1) - D(i)`
`Accordingly, we set up the following table:`
`T(i)   D(i)   D_2(i)1              1 2               1        24               1        37               3        6 13              4       10 23              6       16 39              9        2564             10       35  99             16       51150            15        66216            `
` `
`From the above table, we observe that:`
`D_2(1) = 1, D_2(3) = 3, D_2(5) = 6, D_2(7) = 10, D_2(9) = 15`
`Hence, the odd numbered terms of D_2(i) are triangular numbrs, so that:`
`D_2(11) = 21`
`Again, D_2(2) = 1, D_2(4) = 4, D_2(6) = 9, D_2(8) = 16 `
`Hence, the even numbered terms of D_2(i) are square numbrs, so that:`
`D_2(10) = 25, and D_2(12) = 36`
`Accordingly, we must have:`
`D(11) = D(10) + D_2(10) = 66+25 = 91-> T(12) = T(11) + D(11) = 216+91 = 307`
`D(12) = D(11) + D_2(11) = 91+21= 112-> T(13) = T(12) + D(12) = 307+112= 419`
`D(13 = D(12) + D_2(12) = 112+36= 148-> T(14) = T(13) + D(13) = 419+148= 567`

 Posted by K Sengupta on 2008-12-01 15:16:35

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