Let an be the number of strings (of all lengths) with digits in {1, 2, 3} (repeats allowed) suchthat the sum of the digits in the string is equal to n.(a) Find a recurrence relation satisfied by the an (don't forget the base case(s)).(b) Use your answer to part (a) to prove that an < 2".

We we consider the base cases : n = 1,2,3 first, then deduce the recurrence formula for an

Answered: Let am be the number of strings (of all…

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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2. Let an be the number of ways to write the numbers 1, 2, 3, ..., n in order such that no number i appears immediately before i + 1 (this is a question which appeared on the exam, where an explicit formula was requested!). Explain why an will satisfy the recurrence an = (n-1) an-1+ (n − 2)an-2 for every n ≥ 2.
It is a recurrence relation problem. Please answer both questions as as per your policy you can answer up to 3 questions. But here are only two questions
Solve the recurrence relation an 25an-2. Choose the most appropriate part of the solution from the given options. None of these Or= 5,5 and an = aj. (5)" + a2. 5" Or= 5,-5 and an = a1. (-5)"- a2.5" Or= 5,-5 and an = a1. (-5)" + a2. 5" Next page page
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Let am be the number of strings (of all lengths) with digits in {1, 2, 3} (repeats allowed) such 6. that the sum of the digits in the string is equal to n. (a) Find a recurrence relation satisfied by the an (don't forget the base case(s)). (b) Use your answer to part (a) to prove that an < 2".