Question 31883: Successor Recurrence for Factorial

Original question and givens

Given 7!=5040, use (n+1)!=(n+1)×n! to find 8!.

• G001 — Given 7!=5040, use (n+1)!=(n+1)×n! to find 8!.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-factorial

Step-by-step derivation

source roles = [qf-factorial-successor-recurrence, no-domain-field, successor_recurrence, no-form-field]

derived = [[exact_integer_STRING]]

Factorial requested n ORIGINAL factorial factor2 := mul(1, 2)

Factorial requested n ORIGINAL factorial factor2 = 2

Factorial requested n ORIGINAL factorial factor3 := mul(2, 3)

Factorial requested n ORIGINAL factorial factor3 = 6

Factorial requested n ORIGINAL factorial factor4 := mul(6, 4)

Factorial requested n ORIGINAL factorial factor4 = 24

Factorial requested n ORIGINAL factorial factor5 := mul(24, 5)

Factorial requested n ORIGINAL factorial factor5 = 120

Factorial requested n ORIGINAL factorial factor6 := mul(120, 6)

Factorial requested n ORIGINAL factorial factor6 = 720

Factorial requested n ORIGINAL factorial factor7 := mul(720, 7)

Factorial requested n ORIGINAL factorial factor7 = 5040

source roles = [Factorial requested n, 7]

derived = [5040]

source roles = [7, 5040]

derived = [5040]

Factorial ORIGINAL successor index n plus1 := sum(7, 1)

Factorial ORIGINAL successor index n plus1 = 8

Factorial ORIGINAL successor factor times verified n factorial := mul(8, 5040)

Factorial ORIGINAL successor factor times verified n factorial = 40320

source roles = [7, 5040, 8]

derived = [40320]

Reverse checks

• S012 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S011 · premise_and_result_audit — Recompute Factorial ORIGINAL successor factor times verified n factorial with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Factorial ORIGINAL successor index n plus1 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S008 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.
• S007 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Factorial requested n ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,field roles,domain and premises.Consistency does not imply unique recovery of premises.Keep finite checking,independent review,formal proof and publication separate.

Result

40320

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.