Problem

Evaluate the number of terms in each arithmetic s…

01:38

Problem 22 Hard Difficulty

Evaluate the indicated term for each arithmetic sequence.
$$
a_{10}=-2, a_{15}=-8 ; \quad a_{3}
$$

Answer

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Precalculus

Beginning and Intermediate Algebra

Chapter 14

Sequences and Series

Section 2

Arithmetic Sequences

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Video Transcript

So in this particular problem, we're told we haven't arithmetic sequence where a sub 10 is equal to negative to an ace of 15 is equal to negative. Eight were being asked to find a job. Three. Well, remember, because Ace of 10 is equal to negative to. That tells us that the 10th term in the sequence is negative. Two. And where Ace of 15 is negative. Eight. That tells us that the 15th term in the sequences negative eight. So let's think about this. If I was to write out our sequence, we would know that the 10th term is negative. Two. We don't know the 11th term or the 12th or the 13th or the 14th. However, we do know that the fifth sorry, the 15 term in this sequence is negative. Now, remember, we're talking about an arithmetic sequence, which means to get from the one term to the next we have to add our common value are a constant, which we refer to as our common difference. So in this case, we have to figure out what that common difference would be. So here's one way to do it almost for 10 like this is its own sequence where ace of one is negative two. And in this case, the 123456 term in the sequences Negative eight. So we're gonna use the formula A seven equals ace of one, plus the quantity of and minus one times D in order to find D Now, Like I said, a serve one in this particular case would be negative too. Plus, well, we know that the six term is negative. A so a seven would be negative eight, Which means end would be equal to six. And now we just need to solve this equation for D. Well, six minus one is five. So we're left with negative. Eight is equal to negative two plus five D now to solve for D. We're going to start by adding to to both sides of our equation. So we have negative eight plus two, which is negative. Six is equal to five d and then the Sol for D. We're going to divide both sides of our equation by five. And now we've solved for D so d the common difference is negative. 6/5. Okay, now that we know d what we can do Is we confined the first term in the sequence. Again, we're going to use our formula. A seven is equal to a serve one plus and minus one times D And now we're gonna go back to our original values. I'm going to use the fact that Ace of 10 is equal to negative two. So that means a seven is negative. Two equals. We don't know the first term. It's a someone plus and will begin because it's the 10th term in the sequence and will equal the 10. So love 10 minus one times D, which remember, we just found to be negative. 6/5. Okay, Now we're gonna solve this equation for ace of one. So right have negative two is equal to a sub one. Plus will 10 minus one is nine and nine times negative. 6/5 is equal to negative. 54 5th. So we're gonna have minus negative or sorry, just minus 54 5th. Now, to make this a little bit easier for us, I'm gonna get rid of the fraction by multiplying both sides of our equation by five. Well, five times negative two is negative. 10 and on the right hand side. We have five times a someone, which is five times a serve one and five times negative. 54 5th. It's just negative 54 because the fights will cancel. Now, to solve this equation, we're going to start by adding 54 to both sides. And negative 10 plus 54 is positive. 44. So the salt for eighth of one working the divide both sides by five. So a serve one will equal the 54 Sorry, 44 5th. Well, now that we know D and we know ace of one, remember to get from one term to the next. We just add on our devalue. So in order to find the second term in the sequence, we would take 44 5th and we would add negative 6/5 which is going to equal to 38 5th. And then to find a sub three, we would take the second term 38 5th and again at our devalue negative 6/5 which gives us 32 5th. And now we found the third term. The sequence is 32 5th

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