Prrof By Induction

Question 1

How to show proof by induction of summsation s

Answer 1

1) show true for 1
- summary expression till 1, and sub in 1 to expression, if the same then true

2) assume true for k
So summate till k, and basically replace n with k

3) show true for k +1
- you want expression thst if you sub in k+1 to expression get same thing

• so doing summation to k +1 just means summing to k and the next term, which is the k+1th term
• thus sub in k+1 into the expression to get next term
• in this case if r then just k+1

Now add these together and write in a way so thst it is equal to expression subbed k+1 in it

Use this as aguide

Now proced. Need conclusion

As true for n=k and showed true for n=k+1, then as true for n=1, true for all positive integers greater than 1

Question 2

Make sure to not expand! Same thing as before, just factorise everythign out then expand to mnsimjpulste to what you want

Question 3

Remember that if you have a term higher than a factorial it can combine to just become a higher factorials

Answer 3

So k+2 x k+1 factorial jus beckmes k+2 factorial

Question 4

If can’t expand at the time multiply simplify and then see if can simplify

Question 5

DAMN SOMETIMES IN PROOF TO SHOW INE IS ANITHER, WRITE DOWN EXPECTED, and it won’t factorise out so cleanly in this case what to do

Answer 5

EXPAND EVERYTHING ,

Take out max you can

Now DIVIDE WHAT YOU NEED TO using polynomial

And do the rest

Question 6

Again in factorials., keep dominator same and multiply everything out

Answer 6

And make sure your step 2 is correct lmao

Question 7

Show Matrices That A^K = This

How should you show your matriculated multiplication

Answer 7

Here show it by doing A^k x A^1 first

Do that order and manipulate

Question 8

For matricy gotta make sure that you SHOW THE FUKK VERSION Z(LIKE final substitution)

Answer 8

Do this anyways lol

Question 9

And for sequences what to do

Answer 9

Sub in k to make uk +1 etc