2024 MI PRU2 P1 Q12

In a geometric progression, the first term is \(12\) and the fourth term is \(-\frac{3}{2}\). The sum of the first \(n\) terms of the progression is given by \({{S}_{n}}\) and the sum to infinity of the progression is given by \(S\). Find the least value of \(n\) for which \(\left| S-{{S}_{n}} \right|<0.001\). [5]

An arithmetic progression has the first term \(1000\) and common difference \(-1.5\). Calculate the value of the first negative term of the progression and the sum of all the positive terms. [4]

The sum of the first \(n\) terms of a series is given by \({{S}_{n}}=\frac{1}{3}n\left( n+12 \right)\). By finding an expression for the \({{n}^{\text{th}}}\) term of the series, show that this is an arithmetic series. Hence, state the common difference of the arithmetic series. [4]