Sequences (Induction?) Problem

In summary, the sequence {an} where a1 = sqrt(k), an+1 = sqrt(k + an), and k > 0 is shown to be increasing and bounded in part a. In part b, it is proven that the limit as n approaches infinity of an exists by showing that an is monotonic and bounded. In part c, the limit as n approaches infinity of an is found to be L = \sqrt{k+L} by solving for L.
  • #1
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Homework Statement
Consider the sequence {an} where a1 = sqrt(k), an+1 = sqrt(k + an), and k > 0.

a. Show that {an} is increasing and bounded.

b. Prove that the limit as n approaches infinity of an exists.

c. Find the limit as n approaches infinity of an.

The attempt at a solution

b is straightforward. If you show that an is monotonic and bounded then it has a limit.

I don't really understood how to approach a. The solutions guide suggests some sort of induction that starts with an <= ((1 + sqrt(1 + 4k))/2). I don't understand how I would be able to go from the givens to that point.
 
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  • #2
a) calculate a(n+1)-a(n) and draw your conclusion.

b)-

c) In the limit for n tending to infinity you'll get: [tex]L = \sqrt{k+L} [/tex] which you can solve.
 
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1. What is a sequence?

A sequence is a list of numbers or objects that follow a specific pattern or rule. Each element in a sequence is called a term, and the order of the terms is important.

2. What is the difference between an arithmetic and geometric sequence?

In an arithmetic sequence, each term is found by adding a constant number to the previous term. In a geometric sequence, each term is found by multiplying the previous term by a constant number. In other words, an arithmetic sequence has a constant difference between terms, while a geometric sequence has a constant ratio between terms.

3. How do you find the nth term of a sequence?

To find the nth term of a sequence, you need to identify the pattern or rule that the sequence follows. Once you have the pattern, you can use it to find the missing term by plugging in the value of n.

4. What is the difference between a finite and infinite sequence?

A finite sequence has a limited number of terms, while an infinite sequence continues on indefinitely. In other words, a finite sequence can be written out completely, while an infinite sequence can only be represented by a formula or rule.

5. How is induction used to solve problems involving sequences?

Induction is a mathematical proof technique that is used to prove statements about sequences. It involves showing that a statement is true for the first term in a sequence, and then showing that if it is true for one term, it is also true for the next term. By using induction, we can prove that a statement is true for all terms in a sequence.

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