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Fiud thue 3u4n Ilte follo ning series:Eo 26(5 Marks)Discus the convergemt of the sequence, #hich 0,and 0,+1(5 Marks)...

Question

Fiud thue 3u4n Ilte follo ning series:Eo 26(5 Marks)Discus the convergemt of the sequence, #hich 0,and 0,+1(5 Marks)

Fiud thue 3u4n Ilte follo ning series: Eo 26 (5 Marks) Discus the convergemt of the sequence, #hich 0, and 0,+1 (5 Marks)



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In the sequence $1 !, 2 !, 3 !, 4 !, 5 !, 6 !, \ldots,$ the first term that ends with a zero is $5 ! .$
a. Explain why $5 !$ and all the terms following $5 !$ end with a zero.
b. Find the number of zeros with which $100 !$ ends.

Okay, so we're actually tracked before sequences of arithmetic sequence. So we're gonna track each Terri's common difference and see if they're the same. Do we have our first determines? That's native to minus negative fives, that it's plus five that's equal to three. And then the next two consecutive terms. That's one plus two, which is three and then between one and four. We have for minus one, which is three and then seven and Ford at seven months, for that's equal to three as well. And then from between 10 and seven, we get three as well, so we see that our common difference is the same throughout. So we haven't arithmetic seasons with our common difference e equal to.

All right, So we need to find the 1st 5 terms of this sequence and they give us that first term, which is 24. And so to find the second term, we can plug in a two fer end, then say the second term. Well, if I plug in at two for end right there, in fact, let's just do it by plugging a twofer in two minus one is one. And so that's basically saying that first term and we know that that is 24. So you might as well plug in 24 right now. Hope, and it isn't negative 24. So let's make sure that that is in there over six. And so negative 24 over six is the same thing as saying Negative 24 divided by six, which is a negative, for Let's go ahead and go on to our third term, and I'm gonna do the same thing. I'm gonna start by plugging in this three minus one and recognize that three minus one is too and recognize further, that ace of the second term is negative for so a the second term hey said to the second term is negative for over six. And instead of doing the division, we might as well just make this an equivalent fraction by dividing both the numerator and denominator by two. So we get and you might as well put this negative in front. Doesn't matter if it's on the numerator or in front the same value. There is your third term two months. So let's go ahead and do this fourth term. Things might get a little bit funky cause we're gonna get a fraction of a fraction, but no worries here. So I'm gonna search again by saying, Oh, well, if I plug in that four four minus one is three. And so they're basically saying Plug in this third term in for your numerator there, so this might look a little bit funky, but we have a negative 2/3 in for a numerator in our denominator is a six. And so we're going to rewrite this and say enumerators negative 2/3 and that is multiplied by a 1/6. So this is a dot here to indicate multiplication, and so now we can finish our multiplication. But before we do that, we might as well cross cancel by making this a one. And this is three. You're allowed to do that by dividing both of those by two. Then I'll save you some fraction reduction in the end. Here. One times one is one and six century is 18. Don't forget that there is a negative right here. So that fourth term is negative. 1/18. All right, where we have room. Let's go on this left side here for the fifth term. Just like on the previous ones. I'll go ahead and write. Okay, that that five minus one is there. But that basically means the fourth term, which is right here. That negative 18. So we're gonna rewind and take that away and say Negative and 1/18. Excuse me. I think I said negative. 18 at first. Negative. 1 18 over six. We're gonna rewrite this to have that numerator multiplied by that 1/6. And so we have a one times One is the one. And we have an 18 multiplied by a six, which, if you plug it into your calculator, gives you a 108. Nothing to be reduced year. And as I said before, don't forget that that negative was hanging out in front. That is, there is well, so that fifth term is a negative one. 108th. There's your 1st 5 terms right there. 1231234 and five.

Okay, so I find the first part terms of our sequence given anyone a vehicle to native 24. So it's start by finding h it so a too well. That's equal to a two minus one, which is 8 1/6 for a one that's negative. 24. So we get negative 24/6, which is equal to negative four. Okay, now let's all for our next term. 83 So that's equal to a three minus one, which is 82 over six, and we get a two that was negative for surfacing. Weaken simplified out into native to over three. No, it's all for a four. Well, that's equal to 8 3/6 So a three is negative to over three all over six. So six times three that is 18. So we get negative to over 18 and that can be simplified into what 18 is. Um, can we simplify it out? Let's see. Yeah, that's negative. 1/9. And lastly, or fifth term is a 4/86 our before over six, which is equal to negative 1/9 over six. So that's nine times six, which is equal to negative 1/54

So you have the finite sequence and you want to ride the related Siri's S O. C. This is just some nation off the terms. So you have negative fry. They're going to add that by negative 15 plus 25 plus negatives. 35 plus negatives. 35. So this is the related serious for this sequence here. Sequence. Okay. S o you add everything up on, you should get negative. 125. That so?


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