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 Does it continue? 6: Partial sums (Posted on 2017-09-25)
Before trying the problems "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Part A. Write down the positive integers, cross out every second, and form the partial sums of the remaining.

```1 2 3 4 5 6 7 8 9 10 11
1   4   9   16  25   36
```

Does the pattern of squares continue?

Part B. As before, but cross out every third, form partial sums, then cross out every second and for a second partial sums.

```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 3 7 12 19 27 37 48 61 75 91
1 8 27 64 125 216
```

Does the pattern of cubes continue?

 No Solution Yet Submitted by Jer No Rating

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 Part A | Comment 1 of 7

Yes, it's always a square. The sum of 1,3,5,7,...,2k-1 equals k^2 Proof by induction:

k=1, it's obviously true.

suppose 1+3+5+7+...+2k-1 = k^2

Then the next term is 1+3+5+7+...+(2k-1) + (2k+1) = k^2 + 2k+1 = (k+1)^2

 Posted by chun on 2017-09-25 09:23:17

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