SOLUTION: Write the first five terms of the arithmetic series in which the sum of the second and ninth terms is 25, and the sum of the third and seventh terms is 20.

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Question 1132426: Write the first five terms of the arithmetic series in which the sum of the second and ninth terms is 25, and the sum of the third and seventh terms is 20.
Found 2 solutions by greenestamps, rothauserc:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the 5th term of the sequence; let a be the common difference.

(NOTE: Choosing x to be the 5th term of the sequence puts it halfway between the 3rd and 7th terms; this will simplify the algebra required to solve the problem.)

Then the 2nd term is x-3a; the 9th term is x+4a; the 3rd term is x-2a, and the 7th term is x+2a. The given information is then

%28x-3a%29%2B%28x%2B4a%29+=+25 --> 2x%2Ba+=+25
%28x-2a%29%2B%28x%2B2a%29+=+20 --> 2x+=+20

From those two equations, it is easy to solve for x and a, enabling us to write the first 5 terms of the sequence.

Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
the formula for the nth term of an arithmetic sequence is
:
x(n) = a +d(n-1), where a is the first term, d is the common difference
:
x(2) = a +d(2-1) = a +d
:
x(9) = a +d(9-1) = a +8d
:
x(2)+x(9) = 25 = a +d +a +8d = 2a +9d
:
1) 2a +9d = 25
:
x(3) = a +d(3-1) = a +2d
:
x(7) = a +d(7-1) = a +6d
:
x(3) +x(7) = 20 = a +2d +a +6d = 2a +8d
:
2a +8d = 20
:
2) a +4d = 10
:
solve equation 2 for a
:
a = 10-4d
:
substitute for a in equation 1
:
2(10-4d) +9d = 25
:
20 -8d +9d = 25
:
d = 5
:
a = 10-4(5) = 10 -20 = -10
:
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the first five terms are -10, -5, 0, 5, 10
:
Note
:
x(1) = a +d(1-1) = a = -10
:
x(2) = a +d(2-1) = a +d = -10 +5 = -5
:
x(3) = a +d(3-1) = a +2d = -10 +5(2) = 0
:
X(4) = a +d(4-1) = a +3d = -10 +5(3) = 5
:
x(5) = a +d(5-1) = a +4d = -10 +5(4) = 10
:
check answers
:
x(9) = a +8d = -10 +8(5) = 30
:
x(2) +x(9) = -5 +30 = 25
:
x(7) = a +6d = -10 +6(5) = 20
:
x(3) +x(7) = 0 +20 = 20
:
answer checks
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