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The arithmetic sequence 2, 12, 36, 80.... Is a function of?
A. ${n^2}(n - 1)$
B. $n(n + 1)$
C. ${n^2}(n + 1)$
D. ${n^2}(n + 2)$

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Last updated date: 17th Jun 2024
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Answer
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Hint: This question can be done by putting the value of terms of the sequence in given options and then we can find out the correct option. Basically we have to find the $n^{th}$ term of the sequence.

Complete step-by-step answer:
The arithmetic sequence is given by 2, 12, 36, and 80 here ${T_1} = 2, {T_2} = 12, {T_3} = 36, {T_4} = 80$
Now we will put n=1 in ${n^2}(n - 1)$ and see if the result is coming 2 or not.
$ \Rightarrow {(1)^2}(1 - 1) = 0$
Therefore this function is incorrect as the first term which should be 2 is not coming here.
Now we will put n=1 in $n(n + 1)$ and see if the result is coming 2 or not.
$ \Rightarrow 1(1 + 1) = 2$
Therefore this function can be correct as the first term which should be 2 is coming here but we have to check the other functions as well.
Now we will put n=1 in ${n^2}(n + 1)$ and see if the result is coming 2 or not.
$ \Rightarrow {(1)^2}(1 + 1) = 2$
Therefore this function can be correct as the first term which should be 2 is coming here but we have to check the other functions as well.
Now we will put n=1 in ${n^2}(n + 2)$ and see if the result is coming 2 or not.
$ \Rightarrow {(1)^2}(1 + 2) = 3$
Therefore this function is incorrect as the first term which should be 2 is not coming here.
After this two options are eliminated that are options (A) and (D). Now we will do same process with other two options that are (B) and (C) but now we will put n=2
Now we will put n=2 in $n(n + 1)$ and see if the result is coming 12 or not.
$ \Rightarrow 2(2 + 1) = 6$
Therefore this function is incorrect as the second term which should be 12 is not coming here.
Now we will put n=2 in ${n^2}(n + 1)$ and see if the result is coming 12 or not.
$ \Rightarrow {(2)^2}(2 + 1) = 4 \times 3 = 12$
Therefore this function is correct as required second term 12 is coming here.

So, the correct answer is “Option C”.

Note: Students may likely to make mistake by trying to solve this question by applying direct formula of $n^{th}$ term of an A.P (arithmetic progression) which is given by ${T_n} = a + (n - 1)d$ where a= first term of the sequence and d=common difference which is given by $d = {T_n} - {T_{n - 1}}$. But here the full sequence is not given so the number of terms is not given so this formula cannot be applied directly.