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 Reasonably Rational Sequence (Posted on 2005-06-02)
Fill in the blanks:

2, 2.5, _, 4.25, 5.6, _, _, 9.625, 11.111..., 12.9, ...

 See The Solution Submitted by Dustin Rating: 2.7500 (8 votes)

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 Puzzle Solution Comment 6 of 6 |

Let the ith term of the given sequence be denoted by T(i).

Substituting i = 1,2,3,4,5,...  in turn, we observe that the value of the sequence i*T(i) is: 2, 5,_, 17, 28,.....

Now, 5 = 2+3, and we note that 2 is the 1st prime and 3 is the second prime. Similarly:
17 = 2+3+5+7
28 = 2+3+5+7+11, and so on

Thus we can now assert that, the sequence i*T(i) corresponds to the sum of the first i primes.

We are required to find the 3rd, 6th and 7th terms.

Now,

3*T(3) = The sum of the first three primes is 2+3+5 = 10,
so that: T(3) = 10/3 = 3.33.....

6*T(6) = The sum of the first six primes is 41, so that:
T(6) = 41/6 = 6.833.....
7*T(7) = The sum of the first seven primes is 58, so that:
T(7) = 58/7 = 8.285714.....

 Posted by K Sengupta on 2008-11-04 00:02:42

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