Question

Sol. [Key Concept: $T_{n} = S_{n} - S_{n - 1} = 6n + 2$; sum $\sum_{n = 1}^{10}(6n{+ 2)}^{2}$.]

$${S_{n} = 3n^{2} + 5n }{T_{n} = S_{n} - S_{n - 1} }{\left. \ \Rightarrow T_{n} = \left( 3n^{2} + 5n \right) - \left( 3(n - 1)^{2} \right) + 5(n - 1) \right) }{\Rightarrow T_{n} = 6n + 2 }{\sum_{n = 1}^{10}\mspace{2mu} T_{n}^{2} = \sum_{n = 1}^{10}\mspace{2mu}(6n + 2)^{2} }{= 36\sum_{n = 1}^{10}\mspace{2mu} n^{2} + 24\sum_{n = 1}^{10}\mspace{2mu} n + \sum_{n = 1}^{10}\mspace{2mu} 4 }{= 15220 }$$