This is a discrete math problem. Please explain each step clearly, no cursive writing. **Sequence Definition and Inequality Proof**Define the sequence $\{a_n\}$ as follows:\[a_n = \begin{cases} 1 & \text{for } n = 1, 2, 3 \\a_{n-1} + a_{n-2} + a_{n-3} & \text{for } n \geq 4 \end{cases}\]Prove that for $n \geq 1$, $a_n < 2^n$.

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Description: This is a discrete math problem. Please explain each step clearly, no cursive writing. **Sequence Definition and Inequality Proof**Define the sequence $\{a_n\}$ as follows:\[a_n = \begin{cases} 1 & \text{for } n = 1, 2, 3 \\a_{n-1} + a_{n-2} + a_{n

Given that an=1for n=1.2.3an-1+an-2+an-3for n≥4 The objective is to show that an≤2n for n≥1. Let's…

Answered: 1 for n 1,2,3 An An-1+an-2+an-3 for n…

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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This is a discrete math problem. Please explain each step clearly, no cursive writing.

**Sequence Definition and Inequality Proof**

Define the sequence $\{a_n\}$ as follows:

\[
a_n =
\begin{cases}
1 & \text{for } n = 1, 2, 3 \\
a_{n-1} + a_{n-2} + a_{n-3} & \text{for } n \geq 4
\end{cases}
\]

Prove that for $n \geq 1$, $a_n < 2^n$.

Expert Solution

Step 1

Given that $a_{n} = \{\begin{cases} 1 & f o r n = 1.2.3 \\ a_{n - 1} + a_{n - 2} + a_{n - 3} & f o r n \geq 4 \end{cases}$

The objective is to show that $a_{n} \leq 2^{n}$ for $n \geq 1$ .

Let's prove this by induction,

For $n = 1$

$a_{1} = 1$

Since it is clear that $1 \leq 2$ so $a_{1} \leq 2$ .

Hence, for $n = 1$ , $a_{n} \leq 2^{n}$ is true.

Let's assume that for $n = k$ , $a_{n} \leq 2^{n}$ is true.

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