Amount spent on \[{{1}^{st}}~day\text{ }=\text{ }Rs\text{ }10\]

Amount spent on \[{{2}^{nd}}~day\text{ }=\text{ }Rs\text{ }20\]

Amount spent on \[{{3}^{rd}}~day\text{ }=\text{ }Rs\text{ }40\]

\[10,\text{ }20,\text{ }40,\text{ }\ldots \ldots \]forms a G.P

In which,

\[first\text{ }term,\text{ }a\text{ }=\text{ }10\]

And

\[common\text{ }ratio,\text{ }r\text{ }=\text{ }20/10\text{ }=\text{ }2\text{ }\left( r\text{ }>\text{ }1 \right)\]

The number of days, \[n\text{ }=\text{ }12\]

Thus, the sum of money spend in \[12\text{ }days\] is the sum of \[12\text{ }terms\]of the G.P.

\[{{S}_{n}}~=\text{ }a({{r}^{n~}}-\text{ }1)/\text{ }r\text{ }-\text{ }1\]

\[{{S}_{12}}~=\text{ }\left( 10 \right)({{2}^{12~}}-\text{ }1)/\text{ }2\text{ }-\text{ }1\]

\[=\text{ }10\text{ }({{2}^{12}}-\text{ }1)\text{ }=\text{ }10\text{ }\left( 4096\text{ }-\text{ }1 \right)\]

\[=\text{ }10\text{ }x\text{ }4095\text{ }=\text{ }40950\]

So, the amount spent by him in \[12\text{ }days\text{ }is\text{ }Rs\text{ }40950\]