SOLUTION: An ap has 15 terms and a common difference of -3 .find its first and last terms if their sum is 120.

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Question 1197258: An ap has 15 terms and a common difference of -3 .find its first and last terms if their sum is 120.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
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An AP has 15 terms and a common difference of -3 .
Find its first and last terms if their sum is 120.
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Notice that the 8th term is precisely the middle term of this AP (the term at the center).


Therefore, this mid term doubled is equal the sum of the 1st and 15th terms

    2a%5B8%5D = a%5B1%5D%2Ba%5B15%5D = 120;


Therefore,  a%5B8%5D = 120/2 = 60.


From this point, there is only one step to the completion

    a%5B1%5D =  a%5B8%5D+-+7d = a%5B8%5D+-+7%2A%28-3%29 = 60 + 21 = 81,

    a%5B15%5D = a%5B8%5D+%2B+7d = a%5B8%5D+%2B+7%2A%28-3%29 = 60 - 21 = 39.


ANSWER.  a%5B1%5D = 81,  a%5B15%5D = 39.

Solved and explained, without making unnecessary calculations.


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                 The meaning of this assignment is to teach you to find
        the simplest,  the shortest and the most straightforward way to the completion.



Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The response from the other tutor starts by saying the sum of the first and last terms is 120. While that is probably the intended interpretation of the given information, another possible interpretation is that the sum of all the terms of the ap is 120.

I will show a solution using that interpretation.

The sum is 120, and there are 15 terms, so the average is 120/15 = 8.

In an ap of 15 terms, the average is the one in the middle, which is the 8th term, with 7 terms before it and 7 terms after it.

So the 8th term of the ap is 8.

Then, with a common difference of -3 (that is, the sequence is decreasing), the first term is 8+7(3) = 29 and the last term is 8-7(3) = -13.

ANSWERS: 29 and -13

CHECK: The sum (120) is the number of terms (15) times the average of the first and last terms:

15((29+(-13))/2) = 15(16/2) = 15*8 = 120