Prove the Recursive Sequence Bound Using Mathematical Induction

Learn Mathematical Induction with an important VTU problem involving recursive relations, explained step-by-step. Perfect for BCS405A students to understand recurrence-based proofs and score high in exams. ━━━━━━━━━━━━━━━━━━━━━ 📌 Exact Question: ❓ Given that a(0) = 1, a(1) = 2, a(2) = 3, and a(n) = a(n − 1) + a(n − 2) + a(n − 3), for n ≥ 3, prove that a(n) ≤ 3^n, for all n ∈ ℤ⁺. 📆 Asked in: • Dec 2024 / Jan 2025 – Question 3(b) • Model Question Paper 1 – Question 4(a) 📘 Subject: Discrete Mathematical Structures 📚 Subject Code: BCS405A 📌 Module 2 – Properties of the Integers 📝 Topic: Recursive Relations (Mathematical Induction) ━━━━━━━━━━━━━━━━━━━━━ 🎯 Learning Points: • Recursive definition of sequences • Applying induction to recurrence relations • Bounding sequences using inequalities • Step-by-step inductive reasoning • Solving VTU exam questions efficiently ━━━━━━━━━━━━━━━━━━━━━ 📚 Watch Next: 1️⃣ Next Video (Binomial Theorem Basics) https://youtu.be/TfuzMpQqJPY 2️⃣ Continue Learning https://youtu.be/30VKgKouNmo 3️⃣ More Practice https://youtu.be/pDbb7uq1J88 ━━━━━━━━━━━━━━━━━━━━━ 📘 Module Playlist (Start to Finish): BCS405A Module 2 – Discrete Mathematics https://www.youtube.com/playlist?list=PLM9RnGtTy9_-GqKlk7PZ-O23mSzrDREO3 ━━━━━━━━━━━━━━━━━━━━━ 📙 More Practice / Related Topics: 1️⃣ Strong Induction https://youtu.be/QFXfjJVPepo 2️⃣ Divisibility Proof https://youtu.be/kO_vyEe5Ao8 3️⃣ Inequality Problems https://youtu.be/76vWLgObtZE ━━━━━━━━━━━━━━━━━━━━━ 🚀 Next Step: Move to binomial theorem and coefficient problems. 📌 Don’t stop here: Practice more recursive problems to strengthen your understanding. ━━━━━━━━━━━━━━━━━━━━━ 👉 Follow VTU Maths with Muheeb (Mathematics Tutor) on WhatsApp https://whatsapp.com/channel/0029Vb6ct0L30LKOlXViVx3Y 👉 Get all VTU Maths updates and video links on Telegram https://t.me/vtumathswithmathematicstutor 💎 Support Us: Join our channel and get access to exclusive perks 👇 https://www.youtube.com/channel/UC_OTtwh8_-q-bA7DoS--syg/join ━━━━━━━━━━━━━━━━━━━━━ 📙 All BCS405A Playlists: ▶️ Module 1 – Fundamentals of Logic (Model Papers) https://www.youtube.com/playlist?list=PLM9RnGtTy9_8zuSZb6zBwhnZ8Rxz3fCB1 ▶️ Module 1 – Full Concepts https://youtube.com/playlist?list=PLM9RnGtTy9__4lTRCjni00zPGnoiVIwag ▶️ Module 2 – Mathematical Induction https://youtube.com/playlist?list=PLM9RnGtTy9_-GqKlk7PZ-O23mSzrDREO3 ▶️ Module 3 – Relations and Functions https://youtube.com/playlist?list=PLM9RnGtTy9__YjJR7BBgJl1vqZt5x9PVc ▶️ Module 4 – Recurrence & Inclusion-Exclusion https://youtube.com/playlist?list=PLM9RnGtTy9__-OherPkd9uRdUhXz5ExWJ ▶️ Module 5 – Group Theory https://www.youtube.com/playlist?list=PLM9RnGtTy9_9vSUgbsfNCfn_Hlk8Y20dS ━━━━━━━━━━━━━━━━━━━━━ #DiscreteMathematics #BCS405A #RecurrenceRelation #MathematicalInduction #VTUMaths #EngineeringMaths #VTUExam