SOLUTION: The fifth term of an arithmetic sequence is 9 and the 32nd term is -84. What is the 23rd term?

Question 1043724: The fifth term of an arithmetic sequence is 9 and the 32nd term is -84. What is the 23rd term?
Found 2 solutions by stanbon, rothauserc:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
The fifth term of an arithmetic sequence is 9
a + 8d = 9
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the 32nd term is -84.
a + 31d = -84
------
Subtract and solve for "d"::
23d = -93
d = -4.04
------
Solve for "a"::
a = 9-8*-4.04 = 9+32.32 = 41.32
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What is the 23rd term?
a(23) = 41.32 + 22(-4.04) = -47.64
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Cheers,
Stan H.
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Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
The nth term of an arithmetic sequence is defined as
:
an = a1 + d(n-1), where a1 is the first term and d is the common difference
:
we are given two equations in two unknowns
:
1) 9 = a1 +d(5-1)
2) -84 = a1 +d(32-1)
:
solve equation 1) for a1
:
a1 = 9 -4d
:
substitute for a1 in equation 2)
:
-84 = (9-4d) + 31d
:
27d = -93
:
d = -93/27 = -31/9
a1 = 9 -4(-31/9) = 81/9 +124/9 = 205/9
:
check equation 1) and equation 2)
:
9 = 205/9 + (-31/9)4 = 205/9 - 124/9 = 81/9 = 9
-84 = 205/9 + (-31/9)31 = 205/9 - 961/9 = -756/9 = -84
:
our answers for a1 and d are correct
:
we can calculate the 23rd term
:
a23 = 205/9 + (-31/9)(22) = 205/9 -682/9 = -477/9 = -53
:
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The 23rd term is -53
********************
:

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