SOLUTION: The sixth and eighth terms of an arithmetic sequence are −11 and −19. Find the fifteenth term.

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Question 1178674: The sixth and eighth terms of an arithmetic sequence are
−11 and −19.
Find the fifteenth term.
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
+nth term of an arithmetic sequence is:
a%5Bn%5D=a%5B1%5D%2Bd%28n-1%29

if the sixth and eighth terms of an arithmetic sequence are
a%5B6%5D=-11
a%5B8%5D=-19
we have

-11=a%5B1%5D%2Bd%286-1%29
-11=a%5B1%5D%2B5d.........solve for a%5B1%5D
a%5B1%5D=-11-5d.......eq.1

-19=a%5B1%5D%2Bd%288-1%29
-19=a%5B1%5D%2B7d.........solve for a%5B1%5D
a%5B1%5D=-19-7d.......eq.2
from eq.1 and eq.2 we have
-11-5d=-19-7d......solve for d

7d-5d=-19%2B11
2d=-8
d=-4
go to
a%5B1%5D=-11-5d.......eq.1, substitute d
a%5B1%5D=-11-5%28-4%29
a%5B1%5D=-11%2B20
a%5B1%5D=9

your formula is:
a%5Bn%5D=9-4%28n-1%29


then the a%5B15%5D term is:
a%5B15%5D=9-4%2815-1%29
a%5B15%5D=9-4%2814%29
a%5B15%5D=9-56
a%5B15%5D=-47





Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

The 6th and the 8th terms represent the points -11 and -19 on the number line with the difference of 8 units between them.


There are two gaps of equal length between the 6th and the 8th terms of the AP on the number line.

Hence, the length of each gap is  8%2F2 = 4.



    Since the given AP is decreasing sequence, it implies that the common difference of the AP is -4.



The 15th term is in 7 gaps distance from the 8th term;  hence


    a%5B15%5D = a%5B8%5D - 7*4 = -19 - 28 = -47.


ANSWER.  a%5B15%5D = -47.

From my post, learn how to solve such problem straightforward in short mode.


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    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
    - Chocolate bars and arithmetic progressions
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    - Solved problems on arithmetic progressions
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Also, you have this free of charge online textbook in ALGEBRA-II in this site
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The referred lessons are the part of this online textbook under the topic
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