Induction For INEQUAKITIES

Question 1

How to do
Full step by step, what to do if doesn’t work for baseline first time
What extra to put in conclsuion

Answer 1

Prove true for n= 1 and n= k Hella easy

Prove true for n v k+1 a bit tacky

1) start by doing n = k+1, and splitting using Indices
- as you’re basically multiplying left side by in iy number, multiply right side of the assumed statement by SAME AMOUNT

• now identify target, write it down (by sum bing k+1 into normal)
• now rewrite your left hand side so it LOOKS LIKE TARGET, BUT make adjustments so it’s what it normally should be (so nothing changed)

What you’re tryna do now is say that if the rhs is bigger than the target, then ad the left hand side is bigger than rhs then automatically the lhs is BIGGER THAN TARGET so proved true

As a result you’re just tryna lrofe the extra bit added on is POSITIVE, and thus if positbe then it will be greater

But if it doesn’t work for n= 1 or whatever the baseline was, then might be a logic flaw
NOT TO WORRY, we already proved true for baseline so don’t need to worry, instead make a New baseline
- use next number but to do so must PROVE ITS TRUE
- then use it
And show it works

Finally conclusion

As er use a new baseline, must say as true for n =1 AND N=2 this time

Question 2

How to better prove grewter than 0 for all vslues

Answer 2

Find roots, and draw sketch and thus show for all vslued it s greater

If not convinced, differentiate , and snow grsdient function is ALWAYS increasing for that value, so positive, and thus it’s above 0 for that value

Question 3

If wacky and yiu csn’t make it in the form of the target what to do?

Answer 3

Rewrite the target similar to what you have and see if it is true just facts

If so then done anyways !

Question 4

Proving k + 1 derivative?

Answer 4

That just means derivative of the KTH derivative

And don’t forget we assumed the value for kth derrivstive before, just differneittsr that !

Now remember to get into equal forms might need to us R FORMULA
rearrange and show it fits the TARGET

Question 5

Cool way to do r formula?

Answer 5

Draw triangle amd fill Hypotenuse as R, and oppsite as the value for sin and underneath for value as cos but ntd

Question 6

Remember differentiating? A CONSTSNT k?

Answer 6

Gets RID OF IT!