Sum of the <i>m</i> -th powers of <i>n</i> successive terms of an arithmetic sequence: <i> b <sup>m</sup> </i> + ( <i>a</i> + <i>b</i> ) <i> <sup>m</sup> </i> + (2 <i>a</i> + <i>b</i> ) <i> <sup>m</sup> </i> + … + (( <i>n</i> − 1) <i>a</i> + <i>b</i> ) <i> <sup>m</sup> </i>
Why this work is in the frame
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Bibliographic record
Abstract
This note describes a method for evaluating the sums of the m-th powers of n consecutive terms of a general arithmetic sequence: {Sm ; m = 0, 1, 2, …}. The method is based on the use of a differential operator that is repeatedly applied to a generating function. A known linear recurrence is then obtained and the m-th sum, Sm , is expressed in terms of the preceding ones, Sm −1, Sm −2, … , S 0. This recurrence, which has been derived previously by methods other than the one used here, is solved explicitly for Sm . The final result is expressed in the form of a determinant of order (m + 1) by (m + 1). A comparison is made with other methods, including Inaba's recent approach in this journal.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.011 | 0.011 |
| Meta-epidemiology (narrow) | 0.004 | 0.003 |
| Meta-epidemiology (broad) | 0.007 | 0.002 |
| Bibliometrics | 0.005 | 0.011 |
| Science and technology studies | 0.001 | 0.014 |
| Scholarly communication | 0.001 | 0.005 |
| Open science | 0.014 | 0.004 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it