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Shapes are too Deep (Posted on 2005-06-01) Difficulty: 3 of 5
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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Solution Puzzle Solution Comment 5 of 5 |
(In reply to Answer by K Sengupta)

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
        2
4               1
        3
7               3
        6
13              4
       10
23              6
       16
39              9
       25
64             10
       35 
99             16
       51
150            15
       66
216           
 
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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