Sums of Sequences of Consecutive Squares which are Square
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Theorem
The sums of the following sequences of successive squares are themselves square:
\(\ds \sum_{i \mathop = 7}^{29} k^2\) | \(=\) | \(\ds 7^2 + 8^2 + \cdots + 29^2\) | ||||||||||||
\(\ds \sum_{i \mathop = 7}^{39} k^2\) | \(=\) | \(\ds 7^2 + 8^2 + \cdots + 39^2\) | ||||||||||||
\(\ds \sum_{i \mathop = 7}^{56} k^2\) | \(=\) | \(\ds 7^2 + 8^2 + \cdots + 56^2\) | ||||||||||||
\(\ds \sum_{i \mathop = 7}^{190} k^2\) | \(=\) | \(\ds 7^2 + 8^2 + \cdots + 190^2\) |
Proof
From Sum of Sequence of Squares:
- $\ds \forall n \in \N: \sum_{i \mathop = 1}^n i^2 = \frac {n \paren {n + 1} \paren {2 n + 1} } 6$
Thus:
\(\ds \sum_{i \mathop = 7}^{29} i^2\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{29} i^2 - \sum_{i \mathop = 1}^6 i^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {29 \left({29 + 1}\right) \left({2 \times 29 + 1}\right)} 6 - \frac {6 \left({6 + 1}\right) \left({2 \times 6 + 1}\right)} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {29 \times 30 \times 59} 6 - \frac {6 \times 7 \times 13} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 \, 555 - 91\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 \, 464\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 92^2\) |
\(\ds \sum_{i \mathop = 7}^{39} i^2\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{39} i^2 - \sum_{i \mathop = 1}^6 i^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {39 \left({39 + 1}\right) \left({2 \times 39 + 1}\right)} 6 - \frac {6 \left({6 + 1}\right) \left({2 \times 6 + 1}\right)} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {39 \times 40 \times 79} 6 - \frac {6 \times 7 \times 13} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 20 \, 540 - 91\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 20 \, 449\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 143^2\) |
\(\ds \sum_{i \mathop = 7}^{56} i^2\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{56} i^2 - \sum_{i \mathop = 1}^6 i^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {56 \left({56 + 1}\right) \left({2 \times 56 + 1}\right)} 6 - \frac {6 \left({6 + 1}\right) \left({2 \times 6 + 1}\right)} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {56 \times 57 \times 113} 6 - \frac {6 \times 7 \times 13} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 60 \, 116 - 91\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 60 \, 025\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 245^2\) |
\(\ds \sum_{i \mathop = 7}^{190} i^2\) | \(=\) | \(\ds \sum_{i \mathop = 1}^{190} i^2 - \sum_{i \mathop = 1}^6 i^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {190 \left({190 + 1}\right) \left({2 \times 190 + 1}\right)} 6 - \frac {6 \left({6 + 1}\right) \left({2 \times 6 + 1}\right)} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {190 \times 191 \times 381} 6 - \frac {6 \times 7 \times 13} 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \, 304 \, 415 - 91\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \, 304 \, 324\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 \, 518^2\) |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$