# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Sequences: fixing the general term from an initial condition and a recurrence, and the pitfalls of indexing and finite prefixes](/en/derivations/derivation-mko-sequence-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-sequence)

Let the real sequence a:ℕ→ℝ (with 0∈ℕ) satisfy a₀=5 and a_{n+1}=a_n+3. (1) Prove that exactly one such sequence exists and that a_n=5+3n; (2) compute a₀ to a₄ and a₁₀, explaining that a₀ is the initial term and a₁₀ is the 11th value when counting from 1; (3) write down the finite sequence of length L=4 and its index set; (4) show that the prefix 5, 8, 11 does not determine the sequence.

Sequence

derivation-mko-sequence-01



Concept guide



## [Sequence limits: the exact tail cutoff and an ε-N proof for the reciprocal model](/en/derivations/derivation-mko-sequence-limit-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-sequence)

For u(n)=2−3/(n+1) with n∈ℕ0 (the MKO model with L=2, c=−3, d=1): (1) prove that a candidate cutoff N is valid for ε exactly when 3/(N+1)<ε; (2) find the least valid N for ε=1/2 and explain why N=5 fails; (3) prove u(n)→2 from the ε-N definition; (4) show that N must be chosen after ε.

Sequence limits and exact tail cutoffs

derivation-mko-sequence-limit-01





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