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The sum of first m terms of an A.P. is $4{m^2} - m$. if it's ${n^{th{\text{ }}}}$term is $107$,find the value of n.

Last updated date: 21st Jul 2024
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(Hint: Use the formula of sum of first n terms of A.P. and find first term of A.P. with the help of sum of n terms of A.P.)
The sum of terms is given as,
\[{S_m} = 4{m^2} - m{\text{ }}...{\text{(1)}}\]
Let \[{a_n}\] be \[{n^{th}}\] the term of A.P., then we get,
\[{a_1} = {S_1} = 4{(1)^2} - 1 = 4 - 1 = 3\]
Now, we know that,
\[{S_n} = \dfrac{n}{2}(a + {a_n}){\text{ }}...{\text{(2)}}\]
Also, the value of \[{a_n}\] is given as
\[{a_n} = 107\]
Using the equations and, we get,
\[{S_n} = 4{n^2} - n = \dfrac{n}{2}({a_1} + {a_n})\]
\[4n - 1 = \left( {\dfrac{{3 + 107}}{2}} \right)\]
\[4n - 1 = 55\]
\[n = \dfrac{{56}}{4}\]
\[ \Rightarrow n = 14\]
So, the required solution is \[n = 14\].

Note: In order to solve these types of questions, the first term needs to be calculated first so that the formula for calculating the \[{n^{th}}\]term or the sum, can be applied.