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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 30, 2020
 #1
avatar+965 
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This is just a guess but I think it is -54

 Jul 30, 2020
 #2
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F = First term

D = Common difference

 

9 = F + 4D, 

-84 =F +31D, solve for F, D

F =205 / 9

D = - 31 / 9

 

23rd term =205/9 + (-31/9) * (23-1)

                  =205/9  -(31/9) *22

                  =205/9 - 682/9

                  = - 477 / 9

                  = - 53

 Jul 30, 2020
 #3
avatar+21958 
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The formula for the nth term of an arithmetic sequence is:  tn  =  t1+ (n - 1)d

               tn  =  nth term               t1  =  first term               d   =  common difference

 

fifth term = 9           --->      t5  =  t1 + (5 - 1)d       --->        9  =  t1+ 4d

32nd term = - 84     --->     t32  =  t1 + (32 - 1)d     --->     -84  =  t1 + 31d

 

Subtracting the second equation from the first:  93  =  - 27d     --->     d  =  -93/27

 

Finding the value of t1   --->     9  =  t1+ 4d     --->     9  =  t1 + 4(-93/27)     --->     t1  =  205/9

 

t23  =  205/9 + (23 - 1)(-93/27)     --->     t23  =  205/9 + (22)(-93/27)  =  ........

 Jul 30, 2020

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