Symbiosis National Aptitude Test SNAP 2019 Paper

Let y = log a + log $\left[\frac{{\mathit{a}}^2}{\mathit{b}}\right]$ + log $\left[\frac{{\mathit{a}}^3}{{\mathit{b}}^2}\right]$ + ... + log $\left[\frac{{\mathit{a}}^{\mathit{n}}}{{\mathit{b}}^{\mathit{n}-1}}\right]$
= log a + log ${\mathit{a}}^2$ - log b + log ${\mathit{a}}^3$ - log ${\mathit{a}}^2$ + ... + log ${\mathit{a}}^{\mathit{n}}$ - log ${\mathit{b}}^{\mathit{n}-1}$
= log a + log ${\mathit{a}}^2$ + log ${\mathit{a}}^3$ + ... + log ${\mathit{a}}^{\mathit{n}}$ - [log b + log ${\mathit{b}}^2$ + ... + log ${\mathit{b}}^{\mathit{n}-1}$]
y = (log a + 2 log a + 3log a +...nlog a) – (logb + 2logb + 3logb +...(n -1) log b)
y = (1+ 2 + 3+ ...+ n) log a -(1+ 2 +3 +...n -1) log b
= $\frac{\mathit{n}\left(\mathit{n}+1\right)}{2}$ log a - $\frac{\left(\mathit{n}-1\right)\mathit{n}}{2}$ log b
= $\frac{\mathit{n}}{2}\left[\log {\mathit{a}}^{\left(\mathit{n}+1\right)}-{\mathrm{logb}}^{\left(\mathit{n}-1\right)}\right]$
= $\frac{\mathit{n}}{2}\left[\log \frac{{\mathit{a}}^{\left(\mathit{n}+1\right)}}{{\mathit{b}}^{\left(\mathit{n}-1\right)}}\right]$
y = log ${\left(\frac{{\mathit{a}}^{\mathit{n}+1}}{{\mathit{b}}^{\mathit{n}-1}}\right)}^{\frac{\mathit{n}}{2}}$