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Mathematical Induction: Full and Detailed Tutorials with Several Examples #excellenceacademy — Transcript

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0:00[Music]
0:03all right so let's look at let's look at
0:06mathematical induction all right i said
0:09when it comes to
0:11mathematical induction it's simply a
0:13topic you have to do it you your ability
0:15to want prove mathematical statements
0:18that will cost about mathematical
0:20induction for this topic it's simply
0:22going to prove whether it's statement is
0:25what's true or false nothing more all
0:28right and i said when it comes to muscle
0:30induction you have only three steps
0:32there
0:33number one is called your basic step
0:36for this step what happens anything to
0:38whatever one then check if your left
0:40hand side is equal to what the right
0:43hand side that's your basic step
0:46step three is called the induction step
0:48what was there now any support there
0:51okay
0:52and then we assume it what will be true
0:54abby
0:55they now have the final step and becomes
0:57over there k plus one and i'll show you
0:59how you do that stuff right
1:01let's practice again with this question
1:04so much to service via mathematica prove
1:08that for all
1:10and that this being an element of a real
1:14number prove that this one here now yes
1:17just a usual sample of
1:20uh must be induction question now how do
1:22you solve this fully or basic set
1:24my step one i will say
1:27length and b equals one the basic step
1:30so put x here i have n in my left hand
1:33side
1:33so for the left hand side
1:36i have that one left this
1:38two and minus one now we said let's n
1:42equal to one just because for that su n
1:44of course means two times one a bit
1:47because what that is two times one
1:50that's two and two times one next up
1:53minus one this becomes two times one is
1:56two
1:57minus one whatever now
2:01look out for the right hand side
2:03for right hand side what about here i
2:05have n squared so becomes n squared
2:08instead n is one it becomes n squared
2:11becomes one squared what's one for that
2:15one what the bottom conservation here
2:18left hand side
2:20is all right so i'm done with step one
2:24the sixth step
2:26step two becomes the induction step
2:29for step two inside n support one
2:32okay
2:34and it's gonna be
2:45but of course i'll replace n by k so
2:47because what that
2:49one
2:50plus three
2:52plus five plus dot dot dot
2:55plus two n at the moment
2:58two k n is e
3:00two p
3:02minus one is equal to what that
3:05is where we assume this to be correct
3:10we assume this should be correct
3:13finally step three
3:15um
3:17step three we say
3:19left and equal towards that
3:22eight plus one nah man god
3:24hang on
3:26look up thank god
3:27how do you put enough gifts so what's
3:29the concept there the first person was
3:30actually biggest one i mean yes so say
3:33repeat for first if i
3:35become
3:36one
3:37plus three
3:39plus
3:40five
3:41plus dot dot dot
3:44plus one then
3:47two k minus one then plus one then
3:50you're not equal to n that's what here
3:52what you have here that was the concept
3:54repeat this one first then put the value
3:56of n as you have it yet now mind you
3:59this is what yeah
4:01minus one so put m here as t plus what
4:04you get there this now becomes this look
4:06at this now i'm having two n
4:08first of all the previous one here you
4:10know with that
4:11plus to the power of n here becomes
4:22minus one i have this
4:25it's equal to what that
4:28k plus one over there
4:30all right so this is like the only
4:33uh
4:33thing there right repeat this one here
4:36then for this value of n replace it by
4:39this one of n here by t plus one that's
4:41for your left-hand side for the
4:42right-hand side don't bother about this
4:44one just simply put n as k plus one and
4:46you have this of course squared
4:48this becomes square next of the game we
4:50said
4:51um you can replace it
4:54yeah
4:55so next episode we should replace this
4:57to my last example replace this one here
5:00by this but okay fine
5:03let's do something else i'll call this
5:05one
5:06i'll hold it soon
5:08all right equation one equation two now
5:11look at me
5:12put
5:14equation one
5:15into
5:17equation two how do you solve that topic
5:19from one you're saying that
5:21if i see one plus three plus five plus
5:24dot dot dot plus this all of these is
5:26equal to k square look at this place if
5:29i need to addition two i can see one
5:31plus three this one here plus five here
5:34plus dot dot dot plus this one yeah
5:37let's say that all of these from this
5:38four to this point yeah all of this is
5:40gonna work there
5:42a square so hence i can take out all of
5:45these and this is what they're k squared
5:47instead
5:48take out this mass and call it for that
5:51k square that's all of this here
5:54next up plus
5:55[Music]
5:56what do you guys know
5:582 into
6:00k plus 1
6:03minus 1
6:04equals
6:18all right so simplify this and see it
6:21left hand side is equal to right hand
6:23side that's all
6:24if i simplify this now this becomes
6:28k squared plus from here i'm saying two
6:31multiplied from the bracket
6:33two times d
6:34two k plus two times one
6:37two next up minus one i have it
6:40is equal to k plus one squared means k
6:43plus one is multiplying itself twice
6:49itself
6:50so i have this
6:51so what's up what do you guys here this
6:53gives you
6:55k squared
6:57plus
6:58this gives you what that
6:59should be
7:002 minus 1
7:02that's plus 1 so k squared plus 2k plus
7:062 minus 1 is 1
7:08is equal to absolute values we said use
7:11the first term of perhaps use each of
7:14the tenths in the first bracket multiply
7:17everything in second bracket so take
7:19first step to all of this i have k is
7:23one yeah
7:24eight plus one next part of let's here
7:27plus one that's plus one into one man
7:32three writing one is optional
7:35right from the one is optional
7:37from here now walk this out what do you
7:39guys think this gives you
7:42k square
7:53squared plus k times one k plus one
7:56times k k plus one times one
8:00one i consistently find this is equal to
8:02that
8:03k squared plus two k plus one equal to
8:07what guess is
8:16at this point now you see
8:18since
8:19the left hand side is equal to that
8:22right hand side
8:24therefore the
8:27mathematical statement
8:35is you can say true
8:38or you can see
8:39correct
8:41so that's important about mathematical
8:43induction
8:44proof mathematical statements
8:48question at that time where to be wrong
8:52perhaps but honestly at least your level
8:55give me time for the ones i'm giving you
8:57you are expected to prove it too
8:58all right
9:00yeah sometimes you'll be wrong
9:02like will be different your subconscious
9:04will be different but your level
9:06expected to be one there the same thing
9:09that is constant
9:12[Music]
9:13if you're done
9:15listen
9:17for this anymore to keep going
9:20we can't see it what's the quality we
9:23said for this k plus one this stuff
9:25changes
9:26when it comes to k plus one the first
9:28thing you do is this
9:29go back and repeat this one on your left
9:31hand side
9:33so i'll have this one here yeah that's
9:35it
9:36plus this value of n equals equals one
9:39now look at here i have this one here
9:42so put the value of n here's what then k
9:44plus one so because what they're 2
9:47n what's n then k plus one minus what
9:50that one that's the concept
9:53now hold on
9:54i said this cut this one yeah put it
9:57down again before you do some shot yet
9:59this only works on the left hand side
10:01for the right hand side now you don't do
10:03something
10:04that means you will come here and say
10:05thanks for that yeah and then
10:08just what's there direct substitution
10:10that's it it's a bit technical
10:15all right so use your three steps and
10:16let's if you can solve this one or
10:19prove them to be correct step one we see
10:23legs
10:24and equals one
10:26so look at left hand side and right hand
10:28side so for the left hand side what i
10:32have is just yet
10:33let it come from there
10:35four n
10:36minus one say n is equal to one
10:40so we have four n equal to one so n is
10:42one so because what that
10:44four
10:56[Music]
11:07two times one two n two times one
11:11plus one
11:27listen
11:28listen
11:29um
11:31mix of that vision
11:33means observation whenever you do and
11:36equal to something listen
11:38the numbers you have is always
11:42[Music]
11:54always check so usually let's say i'm
11:56doing a question here on mathematical
11:59induction if i'm saying three over four
12:01here most usually of course obviously
12:04this one yeah three over four one three
12:07four so this number there is always an
12:09indication of what these two look like
12:12when you put n equal to one
12:15that's the concept
12:17the next one gives you
12:19if n is equal to two
12:21right if n is two four times two one
12:23there
12:25huh
12:26minus one
12:27okay
12:28[Music]
12:29so the number you don't guess that
12:31right
12:32don't get me but that's by the way all
12:34right i'm done with first
12:37[Music]
13:05n is it becomes 4k
13:08minus
13:091
13:11is equal to any skill because
13:14into
13:17adding
13:19all right don't forget here's what's
13:20going on there an absorption
13:23and that's not sure next up we say
13:26let anyone to work there
13:29all right so look at this here's that
13:32most
14:01k plus one inside this one here
14:03so because for that
14:06sorry plus right
14:07this
14:08knots
14:09this one here becomes for
14:12c4n
14:14what's that
14:16becomes four n minus one
14:20i'm done with this
14:22is equal to n what's n then
14:26into
14:28n to become
14:43plus one simplify this chord is two
14:45perhaps
14:48how to simplify this you put one into
14:51two
14:51so put
14:53equation one
14:54into two
14:56what does it mean it means absolutely
14:58what they're replacing replacing now
15:02because this
15:03one sorry three plus seven plus eleven
15:07plus four k minus one all of this
15:10is equal to what that
15:12this one
15:14so i will move this one now
15:16and put this one inside so this one goes
15:17off
15:18i'm having this that becomes k into
15:21order
15:222k plus 1. this now represents from here
15:26to here does this
15:28continue with this because what that
15:31plus
15:34point then
15:36[Applause]
15:37k plus one next
15:43[Applause]
15:52next
15:54all right so 2 times k
15:572k
15:582 times 1 2
16:00plus 1 what happens right
16:02simplify this one here how do you start
16:05use this to expand
16:06k plus 2k is what then
16:10k square plus k times 1
16:123
16:13plus works on r so i'm having this as 4
16:17times k
16:19plus 4 times 1
16:214
16:23minus 1
16:25equal to yes 8 plus 1 into let's give
16:28you one here
16:302 plus 1 3 that's great
16:35all right simplify for that uh business
16:38and subtract
16:40right from happy what's here
16:42square
16:43plus k
16:44plus
16:464k into 4 minus
16:49three that's plus three
16:52is equal to this one i want to expand
16:54this one multiply this one on the flat
16:57so i'm having k
16:59into one
17:02next up
17:03plus one optional into order
17:08all right so expand this and simplify as
17:10far as okay
17:12give me what yes
17:13suitcase where
17:15um
17:16plus
17:17three plus peak of bracket we control
17:20that
17:22plus three is equal to k times two k
17:26k square plus k times three is
18:01what does it mean
18:02[Music]
18:04the mathematical
18:14all right it means it's true that's it
18:17yeah
18:18let's do something much more challenging
18:22that's something challenging
18:23[Music]
18:25let's get this done
18:27step one business step
18:28so we say
18:30left hand because we are there
18:33um one check up left hand side
18:36for left hand side
18:38what are there
18:43[Applause]
18:50minus one
18:51that gives you five times one
18:53is five
18:55minus one how many here
18:57i do it wrong
18:59all right next up try out the right hand
19:01side
19:02for right hand side i have that
19:06and about two is equal to what that
19:09all right make n and equal to one so
19:11because what's here
19:13one over two that's n is one
19:16into one yeah
19:17three plus five n that becomes five
19:21times one pi m
19:37what's the division here
19:43is that one dog
19:44[Music]
19:46that one dog
19:50that one
20:02repeat all of this
20:04what you get
20:06four
20:07plus ones
20:08nine
20:09nothing in that
20:11loss
20:12[Music]
20:13loss
20:16loss
20:17five what
20:33all these
20:34equation one
20:36all right we assuming to be correct
20:39all right next up let me see there
20:45three
20:46if n is key plus one
20:48what's the best task
20:50repeat one of it
20:51so he caused one there
20:55not one
20:57plus one
20:59because what then
21:24k plus one minus one
21:27is
21:29[Music]
21:45all right
21:46let me give you
21:48let me give you an observation
21:50let me give you an observation
21:54if you will learn my writing
21:56in some kids
21:57their round bracket is for their square
22:00brackets abby
22:01now look up listen listen
22:04listen
22:07let's say okay
22:09let's say i want to put m in this case
22:11uh let's see key plus one now listen
22:14you now see plus open bracket
22:175 into n what's n there k plus 1 next up
22:22minus 1
22:24this stock is not proper in maths
22:27three it's not quite nice
22:29what do i mean you don't really use two
22:31run brackets in max it's not proper
22:34so what's the concept there if i have a
22:36round bracket what next you got there
22:39square bracket
22:41and after doing that
22:43so no one to change from round bracket
22:46to square bracket
22:47but what if you have a round bracket and
22:49you have square brackets what is it
22:51that's again
23:05again
23:15[Music]
23:49i will come here i will place one there
23:51all of these have it
23:53by what yeah this one so you come from
23:55there
23:58in towards there
24:01this one faces from this point up to
24:04this point next up
24:07so plus
24:12five into one
24:14k plus one
24:18one
24:19is equal to what that
24:21k plus one
24:24into
24:27three plus one
24:29right there
24:32k plus one
24:33so i have
24:35this one here all right
24:37signify this
24:38how does it define this first of all
24:40look at it
24:44this is now reflection one
24:46i was before this fracture what do i
24:48started first
24:50would it be correct if i say this
24:52keyboard everything yes
25:12k times
25:19[Music]
25:21[Applause]
25:29[Music]
25:52so i have this
25:55are we good
25:57all right
25:59christian
26:00what's with you
26:08this one
26:11did you look at the previous one
26:14the same concept now
26:21which one
26:37what we need for now
26:39this is four plus this plus plus this
26:41point here that means by c4 plus nine
26:45plus fourteen plus nineteen
26:48dot dot dot plus this one here
26:50right we set all of this
26:53this one here
26:55is equal to one there this one here you
26:57know
26:59right
27:00from here to here
27:02is equal to this what the constant there
27:04take off this one yeah replace this by
27:07what you have yet that's it
27:10so instead of writing this one here i'll
27:12take it off and put it on instead so it
27:14becomes k plus two down this one here
27:18plus this one again replacing all of
27:20these
27:21i am not left with this one here so
27:23because lost it is this
27:43like this this one like this
27:45huh
27:46if i do talking like this now
27:48zero
27:50zero
27:52let's go wrong now
27:54zero
27:55zero same thing just two months
27:59all right what's happening
28:24[Music]
28:37[Music]
28:41look at me
28:43having this
28:44sorry i said his back again
28:47how do i solve this what do i do here
28:50what's next
28:53here
28:55this is this is where i got questioning
28:58over and over and over again look up
29:00[Music]
29:09look at this
29:28here
29:29is this one here
29:31divided by this
29:3321 over 32
29:36that's seven now because of that
29:38multiplies
29:53that's what
30:21so the problem is this
30:22i'm having yet as one
30:25take else here
30:26f of two and one is what that
30:28two
30:29to become two
30:31into same thing again
30:33swap twenty one over three give us one
30:36sorry 21 about that because what they're
30:38seven this one times this
30:40so two i want to give you one there
30:43one
30:44one times this one there itself
30:48so what do i get
31:00[Applause]
31:03over one this was there
31:05number one what yeah
31:09all right
31:10that's it constantly
31:12having this
31:14it's not equal to look at me
31:19what do i get here
31:24k plus one multiply that
31:43what do i do
31:58so how to expand it you instantly got
32:01that multiplication so because of that
32:04okay
32:05this one finds everything here
32:08five fingers each next up
32:12one get over
32:41is equal to 5 k times 5k
32:46plus k times 8
32:49loss
32:53plus
33:09now listen listen
33:12how do you work on this listen
33:14i will do a rearrangement listen
33:17for this i'll do what that
33:19rearrangement start from what then the
33:21value of kevin was the highest part
33:24please
33:25for this one here one high spot k here
33:28a square so how is that best
33:31so it gives you one there
33:34five k square please check
33:36in
34:32what does that mean
34:56all right
35:15because my magic right
35:19all right
35:20another question
35:23i'm leaving the others
35:26okay
35:27let's go to dinner right
35:29all right so see you
35:30in the next class see you next time
35:34[Music]

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