CAT 2017 Quantitative Ability

Sequence P is an A.P with a = 11, and common difference 3. So, ${\mathit{P}}_{\mathit{k}}=11+\left(\mathit{k}-1\right)3.$
Sequence Q is an A.P with third term 103 and common difference - 4.
${\mathit{t}}_3=\mathit{a}+2\mathit{d}$
$103=\mathit{a}+2\left(-4\right)$ or $\mathit{a}=111$
${\mathit{q}}_{\mathit{k}+2}=111+\left(\mathit{k}+1\right)\left(-4\right)$
${\mathit{q}}_{\mathit{k}+2}=111-4\mathit{k}-4=107-4\mathit{k}$
${\mathit{P}}_{\mathit{k}}>{\mathit{q}}_{\mathit{k}+2}$
$11+\left(\mathit{k}-1\right)3>107-4\mathit{k}$
$8+3\mathit{k}>107-4\mathit{k}$
$7\mathit{k}>99$
$\mathit{k}>\frac{99}{7}$
k has to be an integer, so smallest value k can take is 15