s29=3538 and a29=248

find the A.P

Let a = first term
and d = common difference
Now, a₂₉ = 248
$\Rightarrow$ a + 28d = 248  ............(1)
Sum of first n terms is given by,

$$\begin{gathered} S_n =\frac{n}{2}\left[2a + \left(n - 1\right)d\right] \\ \Rightarrow S_{29} =\frac{29}{2}\left[2a + 28d\right] \\ \Rightarrow 3538 =\frac{29}{2} \times 2\left[a + 14d\right] \\ \Rightarrow \frac{3538}{29} = a + 14d \\ \Rightarrow a + 14d = 122 ..............\left(2\right) \\ \mathrm{Subtracting} \left(2\right) \mathrm{from} \left(1\right), \mathrm{we} \mathrm{get} \\ \left(a + 28d\right) - \left(a + 14d\right) = 248 - 122 \\ \Rightarrow 14d = 126 \\ \Rightarrow \mathit{d}\mathbf{ }\mathbf{=}\mathbf{ }\frac{\mathbf{126}}{\mathbf{14}}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{9} \\ \mathrm{From} \left(1\right), \mathrm{we} \mathrm{have}, \\ a + 28 \times 9 = 248 \\ \Rightarrow a + 252 = 248 \\ \Rightarrow \mathit{a}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{-}\mathbf{4} \\ \mathrm{So}, \mathrm{required} \mathrm{A}.\mathrm{P} \mathrm{is} -4, 5, 14, ..................... \end{gathered}$$