The sum of n terms of an AP is $3 n^{2} + 5 n$ . The $m^{t h}$ term of the AP is 164. Find the value of m. [3 MARKS]

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Solution

Formula: 1 Mark
Application: 2 Marks

We will find an expression for the $n^{t h}$ term. To do that, we will find the difference between the sum of n terms and the sum of (n-1) terms:

$T_{n} = S_{n} - S_{n - 1}$

$= (3 n^{2} + 5 n) - {3 (n - 1)^{2} + 5 (n - 1)}$

$= 3 {n^{2} - (n - 1)^{2}} + 5 {n - (n - 1)}$

$= 3 (2 n - 1) + 5 = 6 n + 2$

Since the $m^{t h}$ term is 164, we have:

$T_{m} = 6 m + 2 = 164$

$\Rightarrow m = 27$

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