Answer:
[tex]a_n=-301 +37(n-1)[/tex]
Step-by-step explanation:
arithmetic sequence formula: [tex]a_n=a +(n-1)d[/tex]
where [tex]a[/tex] is the first term and [tex]d[/tex] is the common difference
Given:
[tex]a_{10}=32[/tex]
⇒ [tex]a +(10-1)d=32[/tex]
⇒ [tex]a+9d=32[/tex]
Given:
[tex]a_{12}=106[/tex]
⇒ [tex]a +(12-1)d=106[/tex]
⇒ [tex]a+11d=106[/tex]
Rearrange the first equation to make [tex]a[/tex] the subject:
a = 32 - 9d
Now substitute into the second equation and solve for [tex]d[/tex]
(32 - 9d) + 11d = 106
⇒ 32 + 2d = 106
⇒ 2d = 106 - 32 = 74
⇒ d = 74 ÷ 2 = 37
Substitute found value of [tex]d[/tex] into the first equation and solve for [tex]a[/tex]:
a + (9 x 37) = 32
a + 333 = 32
a = 32 - 333 = -301
Therefore, the equation is: [tex]a_n=-301 +37(n-1)[/tex]
