SPM Practice Question 5

Question 5 (6 marks):
The sum of the first n terms of an arithmetic progression, S_n is given by $S_n=\frac{3n\left(n-33\right)}{2}.$  
Find
(a) the sum of the first 10 terms,
(b) the first term and the common difference,
(c) the value of q, given that q^th term is the first positive term of the progression.

Solution:
(a)
$\begin{array}{l}{S}_n=\frac{3n\left(n-33\right)}{2}\\ S_{10}=\frac{3\left(10\right)\left(10-33\right)}{2}\\ S_{10}=-345\end{array}$

(b)
$\begin{array}{l}{S}_n=\frac{3n\left(n-33\right)}{2}\\ S_1=\frac{3\left(1\right)\left(1-33\right)}{2}\\ S_1=-48\\ T_1=S_1=-48\\ \\ \text{First term, }a=T_1=-48\\ T_n=S_n-S_{n-1}\\ T_2=S_2-S_1\\ T_2=\frac{3\left(2\right)\left(2-33\right)}{2}-\left(-48\right)\\ T_2=-45\\ \\ \text{Common difference, }d\\ =T_2-T_1\\ =-45-\left(-48\right)\\ =3\end{array}$

(c)
$\begin{array}{l}\text{First positive term, }{T}_q>0\\ T_q>0\\ a+\left(q-1\right)d>0\\ -48+\left(q-1\right)3>0\\ -48+3q-3>0\\ 3q>51\\ q>17\\ \text{Thus, }q=18.\end{array}$