QuestionConsider the following sequence:{an } = {n-n}+=l 1 +x Use Cn+] @n to show that the given sequence {dn is strictly increasing or strictly decreasing:Sequence...

Use the difference $a_{n+1}-a_{n}$ to show that the given sequence $\left\{a_{n}\right\}$ is strictly increasing or strictly decreasing. $$\left\{n-2^{n}\right\}_{n-1}^{+\infty}$$

So we see what this problem is. If we look at the function and e to the negative to end, let's differentiate this with respect to end. So we'll have d d n um and that's gonna ultimately end up giving us, uh, we'll have to dio the product rule so we'll have n times e to the negative to end times a negative, too, plus E to the negative to em. So when we multiple I'll us out and we factor out, we can factor out and eat the negative to end and inside we're gonna have a negative to and plus one. So let's consider this. Now, Um, we noticed that this is the same thing as negative two and plus one over E to the two n. Well, we know that E to the two n is always going to be positive. So what we have is a positive number, and then what we see is that when N is equal to one, we go from one to infinity. When an is equal to one or two or three, we know that it's going to be negative. We're always going to get a negative number. So because the derivative is always negative, we know that the function or the sequence is strictly decreasing.

Problem. Six. The secret has and instance Quit, Benioff end range A strong one this fine and was one s. So that would be envious. One minus one holes Where? So I feel it's taken us minus conference together with one minus a plus one. So one on minus one gets canceled and Blu minus eight has finally difference. So here was one minus. Tm will be four to minus and And this one minus off and minutes. So that's well put up the record. This is minus in squared minus and And when minus was minus, wolves inside it's removed these brackets so minus Boesen sites o minus end Lessons were so this is cancer and we're left with minus tour. So clearly this is negative. So this particular function is strictly do praising

From the number two. The sequence given us minus one while you open from one to infinity. And we have to find whether this is increasing concern Clean, appeasing or crazy. Using the reference test, it was one for minus one who were in this one and will be one minus one $20 difference and was one minus a and B what? Minus one who weren't plus one minus one minus one. So this becomes one minus one over and plus one minus one plus one. Warrant these targets. Cancer. You're lifted. We're here. We have an So the other foot one and minus 11 So this becomes over. And thus one, this will be in plus one minus and cancer for fear. Left this one over and and plus one. So this is the shortness, but it down will always be positive. So this particular sequence is strictly please

Question of the 17. We have to use the differentiation rule to find whether this function or for the CDs and what Windows one has strictly increasing or decreasing. So let's defection this particular function with respect work. The suspect went so the over the n off and over the one plus one as we have used the question. So we have been nominated. Square denominator differentiation off animal. Just be one minus numerator defense station off the one plus one will be plus zero. So just as open of the brackets, we have to win. Plus one minus two women were toe plus one square. So we get these two terms get cancelled. We're left with one over to win this one. No, since the value off our deal where the X is New Orleans office. But do guardian of this particular cities is better than zero. It means that dysfunction is strictly increasing