Use the graphs of the arithmetic sequences {a} and {b} to solve E...

Question

Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58.

Graph of arithmetic sequence with points (1,4), (2,7), (3,10) for exercises 51-58.

If {a} is a finite sequence whose last term is -83, how many terms does {a} contain?

Accepted Answer

Hello everybody A I today that we're going to be talking about this question that states by using the graph of the finite arithmetic sequence. A subscript N determine the numbers of the number of terms it has if its last term is 271. So then they show us a graph here for our sequence where on the Y axis, we have it labeled as a subscript N and the X axis is labeled as N and we have the points one comma 42 comma seven and three comma 10. And the answer choices provided are a 90 B C 87 N D 124. Now, if they tell us that the last term in the last term in our sequence is 270 we have that a subscript N is equal to 271. Well, I'm sorry, they told us that the last term in our sequence was 271. So our, a subscript N will be 271. A subscript N is just going to signify the last term in our sequence. So if we look at our graph, the graph of the arithmetic sequence of a subscript N have the points, one comma four, two comma seven and three comma 10. So we can translate this into a different way of writing it. So one comma four, the X values would be the subscript in our A. So we'll have a of one, if we have one comma four, we have a subscript one is equal to and then the Y value would be the equal to. So a subscript one is equal to four. If we have the 40.1 comma four, if we have the 40.2 comma seven, we have that a of subscript two is equal to the Y value seven. And if we have three comma 10, we have a sub three and that is equal to 10. So now that we have that information, what we wanna do next is something called finding out the common difference. So the common difference and will label it as D is basically, if we subtract these terms, can we get a common difference? So if we have a subscript three minus a subscript two, that is equal to 10 minus seven, which is equal to three. And then we have a subscript two minus a subscript one that is equal to seven minus four, which if you can see the pattern, it'll also be equal to three. And that is why we call it a common difference when we subtract two different terms they have the common answer. So we have that D, the common difference is equal to three. Now, an arithmetic sequence has a certain rule that I will describe now where we have the end the nth term oh And arithmetic sequence with the first term being a one and a common difference. D that nth term would be given by a subscript N is equal to a subscript one plus open parentheses, N minus one closed parentheses, multiplying the common difference D and we have everything we need here. We said earlier that a subscript N is equal to 271. We said that a of subscript one is equal to four and we calculated the common difference D so have that D is equal to three. So now we can go ahead and just plug in. So we'll have, if we plug in, we'll have that 271 which is our, a subscript N, it's equal to four, which is our, a sub one plus open parentheses, N minus one. We don't have an N. So we have N minus one, still closed parentheses and then multiplying an open parentheses three, which is our common difference, closed parentheses. And now we're solving for N which is the NTH term. And that will tell us how many terms there are in our sequence. So now we can distribute the three over. That's multiplying the N minus one. So we'll have 271 is equal to four plus N multiplying the three is three N and then we have negative one or the minus one multiplying the three. So we'll have a minus three or negative three. Now, we can have 271 is equal to four minus three. On the right side is one. So we have three N plus one on the right hand side. And if we move the plus one over to the left by subtracting, we'll have that 270 is equal to three N, which means that now to isolate the end, we divide both sides by three. So we have 270 divided by three, which is 90. So 90 equals N, which means that there will be 90 terms in our sequence which is answer choice. A. So I hope that was helpful. And until next time.

Use the graphs of the arithmetic sequences {a} and {b} to solve Exercises 51-58.
If {a} is a finite sequence whose last term is -83, how many terms does {a} contain?

Key Concepts

Arithmetic Sequence

Finding the nth Term

Finite Sequences

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