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Guess a closed formula for Tn, and prove it.

Guess a closed formula for Tn, and prove it.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1.3. NINETEEN CENTURY Problems 1. "The sum of two consecutive triangular number is a square number". Find a pictorial justification of this statement, and an algebraic proof. 13 2. Represent the reciprocal of a triangular number as the difference of reciprocals of integers. Then use this to find 11/tn. 3. A sequence {n} of real numbers that converges to a number L is said to converge with linear rate of convergence r if 0 < r < 1 and limn→∞ = r. Prove that sequences √a-1 produced with Theon's algorithm converge with linear rate of convergence r = √a+1 Xn+1-L xn-L 4. Modify Theon of Smyrna's system of difference equations to obtain an algorithm for approximating √a for a > 0, and prove that it works. 5. Guess a closed formula for Tn, and prove it.
Transcribed Image Text:1.3. NINETEEN CENTURY Problems 1. "The sum of two consecutive triangular number is a square number". Find a pictorial justification of this statement, and an algebraic proof. 13 2. Represent the reciprocal of a triangular number as the difference of reciprocals of integers. Then use this to find 11/tn. 3. A sequence {n} of real numbers that converges to a number L is said to converge with linear rate of convergence r if 0 < r < 1 and limn→∞ = r. Prove that sequences √a-1 produced with Theon's algorithm converge with linear rate of convergence r = √a+1 Xn+1-L xn-L 4. Modify Theon of Smyrna's system of difference equations to obtain an algorithm for approximating √a for a > 0, and prove that it works. 5. Guess a closed formula for Tn, and prove it.
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