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This question concerns factorials.Let n be a natural number: Which of the following is (are) equal to (n + 1)! ?(n + I)nl(n + 1) .n . (n _ 1) . (n _ 2)..3 . 2("...

Question

This question concerns factorials.Let n be a natural number: Which of the following is (are) equal to (n + 1)! ?(n + I)nl(n + 1) .n . (n _ 1) . (n _ 2)..3 . 2("+(n+2)=(n+l)(n+2)! (n+2)

This question concerns factorials. Let n be a natural number: Which of the following is (are) equal to (n + 1)! ? (n + I)nl (n + 1) .n . (n _ 1) . (n _ 2)..3 . 2 ("+ (n+2)= (n+l) (n+2)! (n+2)



Answers

Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$. $$ 1.2+2.3+3.4+\ldots+n \cdot(n+1)=\left[\frac{n(n+1)(n+2)}{3}\right] $$

In this problem of mathematical induction we have to prove. Given a statement using principle of mathematical induction for all and belongs to national number. First we laid given a statement be our friend. Yeah. And here we have LHs have given a statement one upon one. Multiply by two. Multiply by three plus one upon to multiply by three. Multiply by four plus one upon three. Multiplied by four. Multiplied by five plus up to plus one upon and N plus one. N plus two. And which is equal to end. Multiplying N plus three are born for multiplying in plus one and N plus two. First we prove for P. F one which is basic statement and one is is molest natural number. So first we take a latest part in equal to one putting here. So we have one upon one. Multiply by two. Multiply by three and this will be equal to 1.6. Now we find averages putting an equal to one here, so we have one multiply by four upon four multiply one plus 12 Mhm. Multiplied by three. This is also equal to 1.6. Here we can see alleges equal to rhs so this gets, the statement is true for the given statement. Now they find P R K. Mhm. Up to get the term for the given statement. So we write one upon one. Multiply by two. Multiply by three plus to multiply by three. Multiplied by four plus one upon three. Multiplied by four, multiplied by five plus up to plus one upon Okay, gay plus one multiplying K plus two. Which is acquired. Okay, K plus three whole upon four K plus one. K plus two. Now we need to prove be of K plus one. Yeah. So we take LHs up to K plus one to term life. Be off K. Abdicate some plus. Okay. Plus 100 will be one upon gay plus one. Multiplying K plus two. Multiplying K plus three. So first we need to find these alleges. Mhm. Mhm. Mhm. So we have the R. K. Y. Two K. Multiplying K plus three. All upon four. K plus one. Multiplying K plus two. And this time one upon K plus one. Multiplying K plus two. Multiplying K plus three. So we take common K plus one and K plus two from the denominator. So we have one upon K plus one. Multiplying K plus two within the decade. We have value. Okay K plus three upon four plus one upon K plus three. So taking L. C. M. And simplify the expression. They help one upon K plus one. Multiplying K plus two within brocade, take calcium. So it will be four K. Plus three. So we cross multiply with K plus three here and here. We multiply by four. First. We expand this K. Plus three. Holy square. Mhm. So we write it one upon K. Plus one. Gay plus two. Okay Plus three and four. Also in denominator nominator. We have value. Okay. And expanding the escape plus three. Holy square K. Square plus nine plus six K. Reporting. Mhm. Platform. Mhm. Again we multiply this. Break it with K. Yeah so we have a value equal to we want to do. Mhm. Thank you. Plus nine K. Plus six K. Square plus four. In denominator we have four multiply K. Plus one. Multiply K plus two. Multiplying K. Plus three. Now the met K. Plus one is square mm arranging the nominator. So we write it que. Que. And this time who case Where? Mhm. Okay. Is good plus. Okay. Okay so rest up tom will be for gay square. And this. Mhm. Four cases where plus eight K. And these four will be as it is in the denominator. Yes. We have four multiplying K plus one K plus two and K plus three. We take common gave from the street. Um So we have K. K. Square plus two K plus one. Here we take common for so we have in brigade K square plus two K plus one. Mhm. We didn't wreck aid. We made eight K plus one. Holy square and denominator all terms will be as it is. Yeah. Okay now we take common. K square plus two K plus one. Holy hell. K square plus two K plus one. Multiplying within brigade K plus four. Yeah. Whole upon four K plus one. K plus two. Multiplying K plus three. And we make it K plus one. Holy square. In this case a plus four will be as it is for K plus one. K plus two. Okay plus three. Now we can divide this K plus one by this. K plus one. So we have value. Mhm. K plus one. Also we arranged the escape plus for like K plus one plus three. In denominator we have value. Okay four. We write it. Mhm. This K plus two ways. K plus one plus one. And this K plus three years. A plus one plus two. This is our LHs part. Now we find or ridges putting an equal to K plus one. In the given statement I saw you even the statement N. N plus three. Four. N plus one and N. Plus who? Mhm. So I write it here and and plus three whole upon for and plus one multiplying and plus two here they put in equal to K plus one. So we have. Okay plus one gay plus one plus three. So it will be like a plus one plus three in the denominator we have four gay plus one last one and one more brigade. Okay. Plus one mm plus two. So we can see here religious equal to our riches. Yeah. So we can right here alleges it will do averages and we can say given a statement and if possible is true for all and belongs to national number. And this will be our final answer.

In this problem of principle of mathematical induction we have to prove Given a statement. Using principle of mathematical induction for all and belongs to natural number. We have given a statement. Let your fan thursday eight one multiply to multiply three plus to multiply three. Multiply by four Plus up to plus and and plus one endless two. Which is equal to and N plus one N plus two. N plus three. Whole divide by four. First we prove basic statement be off one, one is the smallest natural number, so we put in equal to one in L. H. Is. So we have a value equal to one. Multiply by two, multiply by three. So it will make well to six. Now we find our riches putting an equal to one here. So we have one multiplied by two, multiplied by three, multiplied by four, divide by four. So it is also equal to six here. Religion is equal to averages. So this basic statement is true. Now we find payoff K for the Cape Town. So it will be well too. In a statement one multiplied by two. Multiply by three. Plus to multiply by three. Multiplied by four up to plus. This will be equal to OK. K plus one. And K plus two in our age is it will be okay, K plus one. K plus two and K plus three. And hold divide by four. Now we have to prove okay plus one is equal to R. A. Chase when N equal to K plus one. So we write it be of gay plus okay +12 term. So it will be well do K plus one. K plus two and gay plus three. And we have value of P. R. K. Which is K. K plus one. K plus two. K plus three, divide by four. Supporting here we have this LHs okay, K plus one. K plus two. K plus three. Holy White by four. And we have K plus one. K plus two and K plus three. Now we take common. So common will be K plus one. K plus two and K plus three in the back. It we helped A by four plus one, solving the brigade. We have value equal to K plus one. K plus two. K plus three. And this will be K plus four and hold the right by four. This is our alleges Now we find our ages putting N equal to K plus one, putting and equal to K plus one. We have value of our chase is equal to Okay plus what? In place of fame and and plus one. So it will be K plus two and plus two. So it will be K plus three. N plus three. So it will be K plus four and hold divide by four. Here, we can see alleges equal to Rh is so we can say given a statement is true for all and belongs to natural numbers. So we write it here, given a statement is true for all and belongs to natural number, and this will be our final answer.

In this problem of mathematical induction we have to prove. Given a statement. Using mathematic principle of mathematical induction for n belongs to natural number. Let given a statement to your fan. Which is Yeah, one multiply by three plus three. Multiplied by five plus five. Multiply by seven plus up to two in minus one. Yeah. Okay. 28 plus one equal to mm hmm. In breaking it for any square plus 16 minus one. Hold up on three. First we prove P off one basic statement here. One is the smallest natural number. So we first taken equal to one. Putting N equal to one. We have LHs. So it will be equal to one multiply by three. So it will be equal to three. Now we find our riches putting N equal to one here. So we have one, four plus six minus one upon three. Simplifying this, we have 10 minus 19 divided by three. So it will be will do three. So we have alleges equal to average is 40 of one. So it is true for the given a statement now we'll find P. R. K. For the Cape Town. And it will be for the given statement one multiply by three plus three. Multiply by five plus five. Multiplied by seven plus up to to k minus one. And second record will be two K plus one and it will be quite okay. Mhm. For Gay square plus six K minus one whole upon three. Now we need to prove the P of K plus one. Yeah. So considering LHs the escape plus K +12 down. So okay plus wanted and will be two K plus one minus one. Multiply by two. Yeah K plus one minus one. We know p of cake will two k four K square plus 16 minus one whole upon three. So we put here okay for gay square the six k minus one. All upon three. And we simplify this package. So it will be will do okay plus two minus one. So it will be well two plus one and these two K plus two. This is plus one. So it will be clear to K plus three. Now we multiply these two breakers. So we have key four K square last six K minus one whole upon three plus, multiplying this to record. We have four K square last six K. Now multiply with one. So it will be okay plus three. Now we simplify this to dance. Taking calcium calcium will be see this time will be as it is four K square plus six K minus one. Plus. This town will be multiplied multiplied by three. So it will be four case girl plus eight K plus three. Now we multiply first working with K. And second brigade with three. So we have four K. Q plus six K. Square minus K. And multiplying second record with three. Well, K square plus 24 K plus night. Hold up on three. After simplifying. It will be well too. Forget you 12 and six case where will be 18 case where minus care in 24 K. It will be 12 to 23 K. And this plus night whole apart three. Now they simplify this record. So did it will be like marriages. We write it for KQ plus 14. Get a squad. He was nine K. Mhm. In four K. Square 14 and four will be 18. Yes 14 K. Hey this night. And hold upon see now we take common K. From first freedoms. So it will be okay. Four case girl. I was 14 K last night taking one comment from this last freedoms. So we have four K. Square plus 14 K. Last night. And hold upon three. Now we take common four K. Square plus 14 K. Plus nine. So we have four K squared plus 14 K. Last night. And this is within the decade K plus one and whole upon three. Yeah. Now we write this barricade simplifying is mm this K plus one as it is. And we write eight. Mhm. For gay square. Yeah plus eight K. Last fall plus six K. Again it will be 14 K. Plus six and minus one. So it will be equal to four plus 16 minus one. Equal to nine. A bar three. No again we simplify so it will be taking four common. So it will be implicated case where plus two K. Plus one. It will be K plus one. Holy square and taking six year comma. So it will be K plus one and this minus one will be as it is. And hold up on three. Now we make it whole you square so it will be four. Okay plus one holy square last six K plus one minus one and whole apart. See no we take carriages putting in equal to K plus one into the given a statement. I show you the given a statement. Given a statement and for any square plus 16 minus one whole upon and so first I write expression and for any square the 16 minus one about 3 14 N equal to K plus one. So it will be K plus one and this will be four K plus one. Holy square last six K plus one minus one. And hold upon three we can see alleges and oranges both are equal. Yeah. Hello jess equal to RHS so we can say given. Mhm. The statement mhm is true. Mhm For on and belongs to national number. And this will be our final answer.

So trying that one plus one plus one. So we'll do this by induction and is gonna be equal. One one plus one is equal. One plus one or two is equal to. So our base case is good inductive case if we let any, okay, we have one plus K is gonna be able to k plus one. So now, to get the K Plus two case, all I have to do is just add one to both sides. You can easily see after we put princes on the terms without switching them around, cause we don't need to use the community property here. Yeah, we can see that this is the same thing for the K plus one term. So our inductive case passes.


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