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Question 35163: Term of a Recurrence with an Initial Value

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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R0069-35163 · q-20260930-r05-sequence-recursive-term-0013

Original question and givens

The real sequence is indexed from 0 with a_0=0 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=10.

  • G001 — The real sequence is indexed from 0 with a_0=0 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=10.

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-recursive-term, [d, n, operation, u]]

After

derived = [recursive_term, no-invented-domain]

Conditions: none

Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract.

Step identifiers and dependencies

Rule: R69_SOURCE_specific_keys_operation_domain_AND_binary_contract · Uses: G001

S002 · implies

Before

source roles = [0, 2, 10, recursive_term]

After

derived = [a_0=u, a_n=u+n*d, zero based]

Conditions: none

Type ALL RAW u/d/count fields EVEN for prefix L=0.Indices start at0 and the formula/recurrence is GIVEN;do not infer a general rule from finite samples.

Step identifiers and dependencies

Rule: Sequence_ALL_RAW_initial_step_and_count_EVEN_empty_prefix · Uses: G001, S001

S003 · calculation

Before

Sequence ORIGINAL recurrence update j=0 := sum(0, 2)

After

Sequence ORIGINAL recurrence update j=0 = 2

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

source roles = [0, 0, 2]

After

derived = [1, 2]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S003

S005 · calculation

Before

Sequence ORIGINAL recurrence update j=1 := sum(2, 2)

After

Sequence ORIGINAL recurrence update j=1 = 4

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

source roles = [1, 2, 2]

After

derived = [2, 4]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S005

S007 · calculation

Before

Sequence ORIGINAL recurrence update j=2 := sum(4, 2)

After

Sequence ORIGINAL recurrence update j=2 = 6

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [2, 4, 2]

After

derived = [3, 6]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S007

S009 · calculation

Before

Sequence ORIGINAL recurrence update j=3 := sum(6, 2)

After

Sequence ORIGINAL recurrence update j=3 = 8

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [3, 6, 2]

After

derived = [4, 8]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S009

S011 · calculation

Before

Sequence ORIGINAL recurrence update j=4 := sum(8, 2)

After

Sequence ORIGINAL recurrence update j=4 = 10

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

source roles = [4, 8, 2]

After

derived = [5, 10]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S011

S013 · calculation

Before

Sequence ORIGINAL recurrence update j=5 := sum(10, 2)

After

Sequence ORIGINAL recurrence update j=5 = 12

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [5, 10, 2]

After

derived = [6, 12]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S013

S015 · calculation

Before

Sequence ORIGINAL recurrence update j=6 := sum(12, 2)

After

Sequence ORIGINAL recurrence update j=6 = 14

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

source roles = [6, 12, 2]

After

derived = [7, 14]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S015

S017 · calculation

Before

Sequence ORIGINAL recurrence update j=7 := sum(14, 2)

After

Sequence ORIGINAL recurrence update j=7 = 16

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

source roles = [7, 14, 2]

After

derived = [8, 16]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S017

S019 · calculation

Before

Sequence ORIGINAL recurrence update j=8 := sum(16, 2)

After

Sequence ORIGINAL recurrence update j=8 = 18

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

source roles = [8, 16, 2]

After

derived = [9, 18]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S019

S021 · calculation

Before

Sequence ORIGINAL recurrence update j=9 := sum(18, 2)

After

Sequence ORIGINAL recurrence update j=9 = 20

Conditions: none

Use ORIGINAL operands exactly;mul/sub/div have EXPLICITLY TWO operands so the old binary primitive cannot ignore a third.Keep RAW SAFE bounds separate from unbounded derived BigInts;do not approximate to force answers.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

source roles = [9, 18, 2]

After

derived = [10, 20]

Conditions: none

At EACH step add the SAME d to the current a_j to obtain a_(j+1),retaining the index.From a0 to an requires EXACTLY n updates.

Step identifiers and dependencies

Rule: Sequence_RECURSIVE_EACH_j_to_jplus1_actual_update · Uses: G001, S021

Reverse checks

  • S022 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S021 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=9 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S019 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=8 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S017 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=7 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S015 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=6 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S013 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=5 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S011 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=4 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S009 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=3 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S007 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=2 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S005 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=1 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S003 · premise_and_result_audit — Recompute Sequence ORIGINAL recurrence update j=0 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.
  • S001 · premise_and_result_audit — Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged.

Result

20

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: 7132e365105295d9c43895e3f60ebb79ec742895d75ccb8374cd09a2268a608e
  • candidate record SHA-256: 6969ea87256551e3c529540816c8437c59190a1f2b7d7e0d5fc6860ee6827774
  • semantic review receipt SHA-256: ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b
  • question-first witness SHA-256: 11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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