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

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

Jul 21, 2017

#1
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9 = F + (5 - 1)*D

-84 = F + (32 - 1)* D, solve for D, F

D = -31/9 common difference

F = 205/9 first term

23rd term = 205/9 + (23 - 1)* (-31/9)

23rd term = -53

Jul 21, 2017
#2
+20831
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The fifth term of an arithmetic sequence is 9 and the 32nd term is -84.

What is the 23rd term?

Formula arithmetic sequence $$a_n = a_1 +(n-1)d$$

$$\begin{array}{|lrcll|} \hline & a_{23} &=& a_1 + (23-1)d \\ (1) & a_{23} &=& a_1 + 22d \\\\ & a_5 = 9 &=& a_1 + (5-1)d \\ (2) & 9 &=& a_1 + 4d \\\\ & a_{32}=-84 &=& a_1 + (32-1)d \\ (3) & -84 &=& a_1 + 31d \\ \hline (1)-(2): & a_{23} - 9 &=& 18d \\ (1)-(3): & a_{23} + 84 &=& -9d \\ \hline & \frac{a_{23} - 9}{a_{23} + 84} &=& \frac{18d}{-9d } \\ & \frac{a_{23} - 9}{a_{23} + 84} &=& -2 \\ & a_{23} - 9 &=& -2(a_{23} + 84) \\ & a_{23} - 9 &=& -2a_{23} -168 \\ & 3a_{23} &=& 9 -168 \\ & 3a_{23} &=& -159 \\ & a_{23} &=& -\frac{159}{3} \\\\ & \mathbf{ a_{23} } & \mathbf{=} & \mathbf{-53} \\ \hline \end{array}$$

Jul 21, 2017