find the 12th term of each sequence. a9= 120, a14=195

The general formula for an Arithmetic Progression is :

                            [tex]\boxed {\boxed {T_n = a_1 + (n-1)d}}[/tex]


[tex]\text {Given that } a_9 = 120,[/tex]

[tex]T_9 = a_1 + (9 - 1)d = 120[/tex]

[tex] a_1 + 8d = 120 \text { ------------------(1)}[/tex]


[tex]\text {Given that } a_{14} = 195,[/tex]

[tex]T_9 = a_1 + (14 - 1)d = 195[/tex]

[tex] a_1 + 13d = 195 \text { ------------------(2)}[/tex]


Equation (2) - Equation (1) :

[tex]13d - 8d = 195 - 120[/tex]

[tex]5d = 75[/tex]

[tex]d = 15[/tex]


Substitute d = 15 into Equation (1) :

[tex]a_1 + 8(15) = 120 [/tex]

[tex]a_1 = 120 - 120[/tex]

[tex]a_1 = 0[/tex]

[tex]\text {Substitute }a_1 = 0 \text{ and d = } 15, [/tex]

               [tex]\boxed { \boxed { T_n = 15(n -1) }}[/tex]


Find 12th term :

[tex]T_{12} = 15(12 -1) = 165[/tex]


[tex]\boxed {\boxed {\text {Answer: The 12th term is 165.}}}[/tex]