Question

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.

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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.