We can see from points
(2,1) means for n = 2, a = 1
(3,3) means for n = 3, a = 3
(4,9) means for n = 4 , a= 9
Now try to see relation in outputs 1,3,9.
We can write 1 as [tex] 3^{0} [/tex]
3 as [tex] 3^{1} [/tex]
9 as [tex] 3^{2} [/tex]
So (2,2) would mean for n =2, a = 1 or [tex] 3^{0} [/tex]----------------(1)
(3,3) would mean for n = 3, a = 3 or [tex] 3^{1} [/tex]-------------(2)
(4,9) would mean for n = 4, a = 9 or [tex] 3^{2} [/tex]--------------------(3)
From (1) we can see for n =2, exponent on 3 is 0
From (2) we can see for n =3, exponent on 3 is 1
From (3) we can see for n =4, exponent on 3 is 2
So we can see the pattern whatever is n value its 2 less is the exponent on 3. So for n exponent on 3 will be n-2
For n = n, a = [tex] 3^{n-2} [/tex]
Now looking at options given
option (A) [tex] \frac{1}{3} 27^{n-1} [/tex] doesnt match to [tex] 3^{n-2} [/tex]
so its incorrect
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option (B) [tex] 27\frac{1}{3}^{n-1} [/tex] doesnt match to [tex] 3^{n-2} [/tex]
so its incorrect
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option (B) [tex] \frac{1}{3} 3^{n-1} [/tex] which we can also write as
[tex] \frac{3^{n-1}}{3} = \frac{3^{n-1}}{3^{1}} = 3^{n-1-1} = 3^{n-2} [/tex]
we subtract exponents when dividing same bases so we subtracted exponent 1 from n-1 and finally got [tex] 3^{n-2} [/tex]
so option (c) matches and is right answer
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option (D) [tex] 3\frac{1}{2}^{n-1} [/tex] doesnt match to [tex] 3^{n-2} [/tex]
so its incorrect
