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# Please help

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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

### 3+0 Answers

#1
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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
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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