Algebraic Proof: The Logic Engine

Master the art of mathematical proof through visual representations and systematic algebraic manipulation

🔬The Proof Engine

Choose a number:

1312

Expression: 2n + 1

n = 1, Value = 3

Odd number: One block doesn't pair

Even number: 2n (has factor of 2)

Odd number: 2n + 1 (even + 1)

Consecutive: n, n+1, n+2

Square number: Write as (...)²

→ Show it's even? Factorise out 2

→ Show it's odd? Get form 2k + 1

→ Show it's a multiple of k? Factorise k

→ Never use x, y, z for consecutive

❌ DON'T: Write "it's obvious" or "clearly true"

❌ DON'T: Skip algebraic steps

❌ DON'T: Use x, y for consecutive integers

✓ DO: Show every manipulation clearly

✓ DO: End with "therefore..." or "hence..."

Pro Tip: In "Show that..." questions, manipulate the LEFT side until it matches the RIGHT side. Your working IS the proof!

Sum of 3 consecutive integers is divisible by 3

(n) + (n+1) + (n+2)

Let the integers be n, n+1, n+2

n + (n+1) + (n+2)

Expand and collect like terms

3n + 3

Factorise

3(n + 1)

Conclusion: Has factor of 3 ✓

Multiple of 3 PROVEN

Bridge Strategy: Phase 1 (Even/Odd visuals) → Phase 2 (Consecutive patterns) → Phase 3 (Complex algebraic proofs)

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