A) \[^{n+1}{{P}_{n+1}}\]
B) \[^{n+1}{{P}_{n+1}}-1\]
C) \[^{n+1}{{P}_{n+1}}-2\]
D) \[^{n+1}{{P}_{n+1}}+1\]
Correct Answer: B
Solution :
\[r\text{th}\] term of the given series \[r{{\cdot }^{r}}{{p}_{r}}=r\cdot r!=\{(r+1)-1\}r!\] \[=(r+1)!-r!\] On putting \[r=1,\,\,2,\,\,3,\,\,...n\] and adding, we get \[^{1}{{p}_{1}}+2{{\cdot }^{2}}{{p}_{2}}+3{{\cdot }^{3}}{{p}_{3}}+...+n{{\cdot }^{n}}{{p}_{n}}\] \[=(2!\,\,-1!)+(3!\,\,-2!)+(4!\,\,-3!)+...+\] \[\{(n+1)!\,\,-n\}\] \[=(n+1)!\,\,-1!\] \[{{=}^{n+1}}{{P}_{n+1}}-1\]You need to login to perform this action.
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