Question

Help Please!

Answer 1

Determine whether the sequence is increasing, decreasing, or not monotonic.

1.\[a_n = \frac{ 4n-8 }{ 8n+5 }\]

2.
\[a_n = 8n+\frac{ 1 }{ n }\]

Answer 2

Take the function \[
f(x)=\frac{4 x-8}{8 x+5}
\]

Answer 3

\[
f'(x)=\frac{84}{(8 x+5)^2}
\]

Answer 4

What can you conclude for the first one?

Answer 5

for the second one do the same
\[
g(x)=8 x +\frac 1 x\]

Answer 6

\[
g'(x)= 8 -\frac 1 {x^2}
\]

Answer 7

What is your conclusion?

Answer 8

Why did you take the derivative?

Answer 9

if the derivative is positive the function will be increasing and so the sequence

Answer 10

If the derivative is negative then it is decreasing

Answer 11

If it is sometimes negative and sometimes positive, then it is not monotone

Answer 12

Are you with me?

Answer 13

Since x=n is always a positive integer, it is clear that both derivatives are positive and hence
the two sequences are increasing