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Question 28777: Exact Error at a Specified Sequence Index

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0056-28777 · q-20261007-advance-20261008-r01-sequence-limit-0077

Original question and givens

For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.

  • G001 — For an arbitrary fixed real L, u(n)=L+(-2)/(n+2) on natural indices starting at zero, with d=2≥1. Find the exact absolute error |u(1)−L|. Give a reduced fraction with positive denominator, retaining /1 for integers.

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-sequence-limit-term-error, sequence_term_error]

After

derived = [exact_absolute_error_fraction]

Conditions: none

Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions.

Step identifiers and dependencies

Rule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001

S002 · implies

Before

source roles = [-2, 2, naturals_from_zero]

After

derived = [2, u(n)=L+c/(n+d), |u(n)-L|=|c|/(n+d)]

Conditions: none

The premise is the WHOLE law for every n>=0,not a finite prefix.With d>0,the denominator is positive and increasing.Subtract L before taking the absolute value.In error/cutoff tasks L is an arbitrary fixed real,not an invented input parameter.

Step identifiers and dependencies

Rule: Sequence_WHOLE_law_and_positive_increasing_denominator · Uses: G001, S001

S003 · calculation

Before

Sequence ORIGINAL n plus d := sum(1, 2)

After

Sequence ORIGINAL n plus d = 3

Conditions: none

Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Sequence ORIGINAL absolute c over n plus d := div(2, 3)

After

Sequence ORIGINAL absolute c over n plus d = 2/3

Conditions: none

Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [-2, 2, 1]

After

derived = [2/3]

Conditions: none

This is |u(n)-L|,so negative c still gives a nonnegative error.Substitute THIS n exactly;do not confuse it with u(n) or a different tail starting point.

Step identifiers and dependencies

Rule: Sequence_exact_error_not_signed_deviation · Uses: G001, S004

Reverse checks

  • S005 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
  • S004 · premise_and_result_audit — Recompute Sequence ORIGINAL absolute c over n plus d with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Sequence ORIGINAL n plus d with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
  • S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.

Result

2/3

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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: a60a1095caa364fc18e33066c58b9ed60fd8a301755195ff83d999a7099ce12a
  • candidate record SHA-256: c2dfcca21585a79f8a041643dc6ae8a822adf445fbc09b50606658715de6a8d0
  • semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8
  • question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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