Find: $\;$ $21 + 18 + 15 + \cdots - 81$
For the given arithmetic progression (A.P) $\;$ $21 + 18 + 15 + \cdots - 81$
first term $= a = 21$
common difference $= d = -3$
Let the number of terms in the given A.P $= n$
$n^{th}$ term $= t_n = -81$
$t_n = a + \left(n - 1\right)d$
i.e. $\;$ $-81 = 21 + \left(n - 1\right) \times \left(-3\right)$
i.e. $\;$ $n - 1 = \dfrac{21 + 81}{3} = 34$
$\implies$ $n = 35$
For the given A.P, sum to $n$ terms is
$\begin{aligned}
S_n & = 21 + 18 + 15 + \cdots - 81 \\\\
& = \dfrac{n}{2} \left[2a + \left(n - 1\right)d\right] \\\\
& = \dfrac{35}{2} \left[2 \times 21 + \left(35 - 1\right) \times \left(-3\right)\right] \\\\
& = \dfrac{35}{2} \left[42 - 102\right] \\\\
& = \dfrac{35}{2} \times \left(-60\right) = -1050
\end{aligned}$