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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 2, 2018

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

AP Formula:

$$\begin{array}{|rcll|} \hline a_i(j-k)+a_j(k-i)+a_k(i-j) &=& 0 \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline a_5 &=& 9 \qquad & i = 5 \\ a_{32} &=& -84 \qquad & j = 32 \\ a_{23} &=& \ ? \qquad & k = 23 \\\\ a_i(j-k)+a_j(k-i)+a_k(i-j) &=& 0 \\\\ a_5(32-23) + a_{32}(23-5) +a_{23}(5-32) &=& 0 \\ 9\cdot 9 - 84 \cdot 18 - a_{23}\cdot 27 &=& 0 \\ a_{23}\cdot 27 &=& 9\cdot 9 - 84 \cdot 18 \\ a_{23}\cdot 27 &=& 81 - 1512 \\ a_{23}\cdot 27 &=& - 1431 \\\\ a_{23} &=& -\dfrac{ 1431 } {27} \\\\ \mathbf{ a_{23} } & \mathbf{=} & \mathbf{ -53 } \\ \hline \end{array}$$

The 23rd term is -53

Jul 2, 2018