sequence 2

**kastamonu** · February 25th 2013, 06:47 AM

General term in a sequence is a_n=4n+8/(n+5)
What is the sum of greatest lower bound of the sequence and lowest upper bound f the sequence. I got the answer as 5 but it must be 6.

**Plato** · February 25th 2013, 06:58 AM

Originally Posted by kastamonu

General term in a sequence is a_n=4n+8/(n+5)
What is the sum of greatest lower bound of the sequence and lowest upper bound f the sequence. I got the answer as 5 but it must be 6.

If $n\in\mathbb{Z}^+$ then $2\le\frac{4n+8}{n+5}\le 4$ .

**kastamonu** · February 25th 2013, 07:15 AM

How did you find it?

**MINOANMAN** · February 25th 2013, 07:26 AM

VERY EASY...FIND the first term a1=...2 and then the lim of the sequence =4.....then think...it is easy.

**kastamonu** · February 25th 2013, 07:32 AM

I tried to find the answer by evaluating the fraction. Yes that way is better. Limit is 4.

**kastamonu** · February 25th 2013, 10:05 AM

4n+8 4n+20-20+8
---- = ----------
n+5 n+5

12
= 4 - ---
n+5 .

Sequence is increasing and lower bound is 2(If we take n=1). If we
find the limit according to infinity upper bound is 4. But I wanted to
find it by using inequality method.

**HallsofIvy** · February 25th 2013, 11:02 AM

$\frac{4n+8}{n+5}\le A$ is, since n+ 5 is positive, the same as $4n+ 8\le A(n+ 5)= An+ 5A$ . $(A- 4)n\ge 8- 5A$ . If we want this to be true for all n, we must have A= 4.

**kastamonu** · February 25th 2013, 10:54 PM

many thanks

**Prove It** · February 25th 2013, 11:39 PM

Originally Posted by kastamonu

General term in a sequence is a_n=4n+8/(n+5)
What is the sum of greatest lower bound of the sequence and lowest upper bound f the sequence. I got the answer as 5 but it must be 6.

$\displaystyle \begin{align*} \lim_{n \to \infty} \frac{4n + 8}{n + 5} &= \lim_{n \to \infty} \frac{4n + 20 - 12}{n + 5} \\ &= \lim_{n \to \infty} \left[ \frac{4(n + 5)}{n + 5} - \frac{12}{n + 5} \right] \\ &= \lim_{n \to \infty} \left( 4 - \frac{12}{n + 5} \right) \\ &= 4 - 0 \\ &= 4 \end{align*}$

So the least upper bound is 4.

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