50th term of the sequence 16,48,96,160,_____ is ?

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$$\begin{gathered} 16,48,96,160,.... \\ 48-16=32 \\ 96-48=48 \\ 160-96=64 \\ \mathrm{which} \mathrm{means} \mathrm{difference} \mathrm{of} \mathrm{terms} \mathrm{is} \mathrm{in} \mathrm{A}.\mathrm{P} \mathrm{with} \mathrm{common} \mathrm{difference} \\ \mathrm{i}.\mathrm{e}. \mathrm{d}=16 \mathrm{and} \mathrm{first} \mathrm{term}=\mathrm{a}=32 \\ \mathrm{How} \mathrm{do} \mathrm{we} \mathrm{third} \mathrm{term}? \\ \mathrm{Third} \mathrm{term}=\mathrm{First} \mathrm{Term}+\left(\mathrm{Sum} \mathrm{of} \mathrm{terms} \mathrm{upto} \mathrm{third} \mathrm{term}\right) \\ \mathrm{i}.\mathrm{e}. \\ 96=16+\left(32+48\right) \\ \mathrm{How} \mathrm{do} \mathrm{we} \mathrm{get} \mathrm{fourth} \mathrm{term}? \\ \mathrm{Fourth}=\mathrm{First} \mathrm{Term}+\left(\mathrm{Sum} \mathrm{of} \mathrm{differences} \mathrm{upto} \mathrm{fourth} \mathrm{term}\right) \\ 160=16+\left(32+48+64\right) \\ \mathrm{Similarly} \\ {\mathrm{T}}_{50}=\mathrm{First} \mathrm{term}+\left(\mathrm{Sum} \mathrm{of} \mathrm{differences} \mathrm{upto} 49\mathrm{th} \mathrm{term}\right) \\ =16+\left(32+48+64+.....49\mathrm{th} \mathrm{term}\right) \\ =16+\frac{49}{2}\left[2\mathrm{a}+\left(49-1\right)\mathrm{d}\right] \left[{\mathrm{Sum}}_{\mathrm{A}.\mathrm{P}.}=\frac{\mathrm{n}}{2}\left\{2\mathrm{a}+\left(\mathrm{n}-1\right)\mathrm{d}\right\}\right] \\ =16+\frac{49}{2}\left[2\times 32+48\times 16\right] \\ =16+\frac{49}{2}\left[832\right] \\ =16+20384 \\ \mathbf{=}\mathbf{20400}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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