(b)From part (a),we know that lim an exists. Compute it.n→∞ 4. Consider a sequence aj = 3 and an+1 = vV12 + an for each n > 1.(a)by 4.Show by induction that {an} is monotone increasing and bounded above

Given that a1=3 and an+1=12+an for each n≥1 Also, the sequence an is monotonic increasing and…

Answered: (b) From part (a), we know that lim an…

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(b) From part (a), we know that lim an exists. Compute it. n00

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 41E

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

(b)
From part (a),
we know that lim an exists. Compute it.
n→∞

4. Consider a sequence aj = 3 and an+1 = v
V12 + an for each n > 1.
(a)
by 4.
Show by induction that {an} is monotone increasing and bounded above

Expert Solution

Step 1

Given that $a_{1} = 3 a n d a_{n + 1} = \sqrt{12 + a_{n}}$ for each $n \geq 1$

Also, the sequence $\{a_{n}\}$ is monotonic increasing and bounded above by 4.

This implies that the sequence $\{a_{n}\}$ is convergent. Also, the sequence is the sequence of positive terms.

The objective is to find $\lim_{n \to \infty} a_{n}$

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