+0  
 
0
65
2
avatar+147 

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

 Dec 5, 2018
 #1
avatar+3613 
+1

\(a_n = a_1 + (n-1)d\\ a_5 = a_1 + 4d = 9\\ a_{32} = a_1 + 31d = -84\\ \text{subtract }a_5 \text{ from }a_{32}\\ 27d=-93\\ d=-\dfrac{93}{27} = -\dfrac{31}{9}\\ a_1 = 9-4d = 9-4\left(-\dfrac{31}{9}\right) = \\ 9+\dfrac{124}{9} = \dfrac{205}{9}\\ a_{23} = a_1 + 22d = \dfrac{205}{9}+22\left(-\dfrac{31}{9}\right) = -53 \)

.
 Dec 5, 2018
 #2
avatar+20837 
+6

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

 

Formula of an arithmetic sequence:

\(\begin{array}{|rcll|} \hline \begin{vmatrix} a_i & a_j & a_k \\ i & j & k \\ 1 & 1 & 1 & \\ \end{vmatrix} = 0 \\\\ a_i(j-k)+a_j(k-i)+a_k(i-j) &=& 0 \\ \hline \end{array} \)

 

\(\text{Set $i=5$, $\ j=32$, $\ k=23$ } \\ \text{Set $a_i=a_5=9$, $\ a_j=a_{32}=-84$, $\ a_k=a_{23}$ } \)

 

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

 

laugh

 Dec 6, 2018

8 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.