SOLUTION: The sum of the first ten terms of a linear sequence is -60 and the sum of the first fifteen terms of the sequence is -165.find the 18th term of the sequence

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Question 1150834: The sum of the first ten terms of a linear sequence is -60 and the sum of the first fifteen terms of the sequence is -165.find the 18th term of the sequence
Found 3 solutions by greenestamps, MathLover1, MathTherapy:
Answer by greenestamps(13209) About Me  (Show Source):
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Let a = first term
Let d = common difference

10th term: a+9d
15th term: a+14d
18th term: a+17d

Sum of first ten terms: -60 = 10 times average of first and tenth terms

10%28%28a%2B%28a%2B9d%29%29%2F2%29=+-60
2a%2B9d+=+-12 [1]

Sum of first 15 terms: -165 = 15 times average of first and 15th terms 15%28%28a%2B%28a%2B14d%29%29%2F2%29+=+-165
2a%2B14d+=+-22 [2]

Subtract [1] from [2] and solve for d; then use that value in either [1] or [2] to solve for a.

Finally, use the values of a and d to find the 18th term.

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
linear sequences are sequences where the difference between successive terms is always the same
In General we could write an arithmetic sequence like this:
a, a%2Bd, a%2B2d, a%2B3d, ...

The sum of the first n terms of an arithmetic sequence use this formula:

S%5Bn%5D=%28n%2F2%29%282a%2B%28n-1%29d%29 where a is the first term, n the number of terms, and d is common difference

given:
The sum of the first ten terms of a linear sequence is -60+; so, we have
-60=%28n%2F2%29%282a%2B%28n-1%29d%29........... ten terms=>n=10
-60=%2810%2F2%29%282a%2B%2810-1%29d%29
-60=5%282a%2B9d%29
-60%2F5=%282a%2B9d%29
-12=2a%2B9d
2a=9d%2B12
a+=+-%289+d%29%2F2+-+6..................eq.1

and the sum of the first fifteen terms of the sequence is -165
-165=%28n%2F2%29%282a%2B%28n-1%29d%29........... 15 terms=>n=15
-165=%2815%2F2%29%282a%2B%2815-1%29d%29
-330%2F15=2a%2B14d
-22=2a%2B14d
-11=a%2B7d
a=-7d-11.................eq.2

from eq.1 and eq.2 we have

-%289+d%29%2F2+-+6=-7d-11
%285+d%29%2F2+=-+5+
5d=-10
d=-2

find first term:
a=-7d-11.................eq.2
a=-7%28-2%29-11
a=14-11
a=3

find the 18th term of the sequence:
n=18, a%5B1%5D=3, and d=-2
a%5B18%5D=3%2B%2818-1%29%28-2%29
a%5B18%5D=3%2B17%28-2%29
a%5B18%5D=3-34
a%5B18%5D=-31

so, your sequence is:
3,1,-1,-3,-5,-7,-9,-11,-13,-15,-17,-19,-21,-23,-25,-27,-29,-31,....




Answer by MathTherapy(10556) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of the first ten terms of a linear sequence is -60 and the sum of the first fifteen terms of the sequence is -165.find the 18th term of the sequence
Sum of "n" terms of an A.P.: matrix%281%2C3%2C+S%5Bn%5D%2C+%22=%22%2C+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29%29
matrix%281%2C3%2C+S%5B10%5D%2C+%22=%22%2C+%2810%2F2%29%282a%5B1%5D+%2B+%2810+-+1%29d%29%29 ------- Substituting 10 for n
 ------ Substituting - 60 for S%5B10%5D

matrix%281%2C3%2C+S%5B15%5D%2C+%22=%22%2C+%2815%2F2%29%282a%5B1%5D+%2B+%2815+-+1%29d%29%29 ------- Substituting 15 for n
matrix%281%2C3%2C+-+165%2C+%22=%22%2C+15%282a%5B1%5D+%2B+14d%29%2F2%29 ------------- Substituting - 165 for S%5B15%5D

5d = - 10 ------- Subtracting eq (i) from eq (ii)
d, or common difference = matrix%281%2C3%2C+%28-+10%29%2F5%2C+%22=%22%2C+-+2%29

matrix%281%2C3%2C+2a%5B1%5D+%2B+9%28-+2%29%2C+%22=%22%2C+-+12%29 ------ Substituting - 2 for d in eq (i)

To find a%5B18%5D, we substitute  into the formula for a term in an A.P., as follows: