1. ## sequence

Hi, I'm not sure how to figure out this problem:

Let
A. Find the smallest real number such that is decreasing for all greater than

B. Find the smallest integer such that is decreasing for all integers greater than or equal to .

Also, I need help with finding the equation $A_{n}$ for this sequence: $\frac{3}{16},\frac{4}{25},\frac{5}{36},\frac{6}{49 }$ I Know that there is going to be n+2 on the numerator but I can't figure out the denominator, so its $\frac{n+2}{x}$ and I need help finding x.
Regards,
Matt

2. Originally Posted by matt3D
Hi, I'm not sure how to figure out this problem:

Let
A. Find the smallest real number such that is decreasing for all greater than

B. Find the smallest integer such that is decreasing for all integers greater than or equal to .
$f(x)$ is decreasing where $f'(x) < 0$

$f(x)$ is increasing where $f'(x) > 0$

Also, I need help with finding the equation $A_{n}$ for this sequence: $\frac{3}{16},\frac{4}{25},\frac{5}{36},\frac{6}{49 }$ I Know that there is going to be n+2 on the numerator but I can't figure out the denominator, so its $\frac{n+2}{x}$ and I need help finding x.
Regards,
Matt
notice that the denominator are all perfect squares.

3. Hmm, okay, so $f'(x)=\frac{-(x^2-57)}{(x^2+8x+57)^2}$
and I do see that they are perfect squares, so then it should be something like n to a power or something?

4. Originally Posted by matt3D
Hmm, okay, so $f'(x)=\frac{-(x^2-57)}{(x^2+8x+57)^2}$
yes. now for which values of x is this positive? how about negative?

and I do see that they are perfect squares, so then it should be something like n to a power or something?
note that the first denominator is $4^2$, the second is $5^2$, the third is $6^2$, the fourth is $7^2$

5. Okay, so it is positive when $x=-\sqrt{57}$ and negative from $-\infty\ to\ \sqrt{57}$ and is undefined at $\sqrt{57}$ I see where you are coming from for the sequence but I still can't figure out an equation for it.

6. Okay, I got the equation: $\frac{n+2}{(3+n)^2}$

7. Originally Posted by matt3D
Okay, so it is positive when $x=-\sqrt{57}$ and negative from $-\infty\ to\ \sqrt{57}$
no. check again. note that the numerator is the difference of two squares and the denominator is always positive

and is undefined at $\sqrt{57}$
the function and its derivative are defined everywhere.

I see where you are coming from for the sequence but I still can't figure out an equation for it.
you figured out a formula for the numerators. why is this so hard for you. you have increasing integers, each squared. you were already able to find a sequence to give increasing consecutive integers.

8. Originally Posted by matt3D
Okay, I got the equation: $\frac{n+2}{(3+n)^2}$
yes, for n = 1,2,3,4, ...

and you should probably call the terms something, like $a_n$

9. Originally Posted by Jhevon
no. check again. note that the numerator is the difference of two squares and the denominator is always positive

the function and its derivative are defined everywhere.

you figured out a formula for the numerators. why is this so hard for you. you have increasing integers, each squared. you were already able to find a sequence to give increasing consecutive integers.
Okay, so what are you implying by saying that the denominator is always positive? I really appreciate your help. I know that $x=\sqrt{57}\ and\ x=-\sqrt{57}$

10. Originally Posted by matt3D
Okay, so what are you implying by saying that the denominator is always positive? I really appreciate your help. I know that $x=\sqrt{57}\ and\ x=-\sqrt{57}$
yes, since the denominator is always positive, it means we change sign according to the numerator. that is, if the numerator of the derivative is positive, then the derivative is positive. if it is negative, then the denominator is negative. so find the intervals on which the numerator is positive and on the interval where is is negative. that will tell you where your function is increasing or decreasing and you can answer the question