Q.Find the sum of the first $25$ terms of the Arithmetic Progression $2, 5, 8, 11, \ldots$

Use the AP sum formula $S_n = \frac{n}{2}[2a + (n-1)d]$.

Here $a = 2$, $d = 5 - 2 = 3$ (checked: $8 - 5 = 3$, $11 - 8 = 3$), $n = 25$.

Apply $S_n = \dfrac{n}{2}\big[2a + (n-1)d\big]$:

$S_{25} = \frac{25}{2}\big[2(2) + (25-1)(3)\big] = \frac{25}{2}\big[4 + 72\big] = \frac{25}{2}(76) = 25 \times 38 = 950.$ …

Pairing method: the last term is $74$; pair the 1st with the 25th ($2 + 74 = 76$), and note the 13th (middle) term is $t_{13} = 2 + 12(3) = 38$. There are $12$ pairs each summing to $76$ plus …

Using $n$ instead of $(n-1)$ inside the bracket gives an extra $d$: the wrong $[2(2) + 25(3)] = 79$ leads to $S_{25} = \tfrac{25}{2}(79) = 987.5$, which …