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

We  have  that

a₃₂  = a₅  + common difference * [ 32 - 5]

-84  = 9  + common difference * [ 27]        subtract  9 from both sides

-93 = common difference * [27]              divide  both sides  by 27

-93/27 =  common difference   = -31/9   

a₂₃  =  a5  + (-31/9) (23 - 5)

a₂₃  =  9  + (-31/9) (18) 

a₂₃ =  9  - 62

a₂₃ = -53