Solution/s:
Given:
[tex]n = 24[/tex]
[tex]a_2 = 6[/tex]
[tex]a_5 = 27[/tex]
[tex]a_2 = a_1 + d[/tex]
[tex]a_5 = a_1 + 4d[/tex]
Let us find [tex]d[/tex]:
[tex] a_5 = a_1 + 4d [/tex]
Substitute [tex] a_2 [/tex]
[tex] a_5 = (a_1 + d) + 3d [/tex]
[tex]a_5 = a_2 + 3d[/tex]
Substitute 6 and 27:
[tex](27) = (6) + 3d[/tex]
[tex]27 - 6 = 3d[/tex]
[tex]21 = 3d[/tex]
[tex]7 = d[/tex]
Now we can find [tex]a_1[/tex]:
[tex] a_2 = a_1 + d[/tex]
[tex] 6 = a_1 + 7 [/tex]
[tex] a_1 = 6 - 7 [/tex]
[tex] a_1 = -1 [/tex]
Now we can find the general term:
[tex] a_n = a_1 + (n - 1)d [/tex]
[tex] a_n = -1 + (n - 1)7 [/tex]
[tex] a_n = -1 + 7n - 7 [/tex]
[tex] a_n = 7n - 8 [/tex]
Now we can find the 24th term:
[tex] a_{24} = 7(24) - 8 [/tex]
[tex] a_{24} = 168 - 8 [/tex]
[tex] a_{24} = 160 [/tex]