2022 SJI P1 Q7

Consider an arithmetic series

\[\ln x+\ln(x^a)+\ln(\sqrt x)+\ldots\quad\text{where}\hspace{0.5em}x\in\mathbb R,\ x>1\text{ and}\hspace{0.5em}a\in\mathbb R,\ a\neq0.\]

1. Find the value of \(a\).[3]
2. Find the common difference of the series in the form \(d\ln x\) where \(d\in\mathbb Q\).[2]
3. The sum of the first \(n\) terms of the series is equal to \(\ln(x^{-4.5})\). Find the value of \(n\).[5]

Consider the geometric series

\[\lg x+\lg(x^b)+\lg(\sqrt x)+\ldots\quad\text{where}\hspace{0.5em}x\in\mathbb R,\ x>1\text{ and}\hspace{0.5em}b\in\mathbb R,\ b\neq0.\]

4. Find the possible values of \(b\).[5]
5. The sum to infinity of the above series is equal to \(2+\sqrt2\) when \(b>0\). Find the value of \(x\).[5]