A) \[{{(1+x)}^{n}}>(1+nx),\] For all natural numbers n
B) \[{{(1+x)}^{n}}\ge (1+nx),\] For all natural numbers n, Where \[x>-1\]
C) \[{{(1+x)}^{n}}\le (1+nx),\] For all natural numbers n
D) \[{{(1+x)}^{n}}<(1+nx),\] For all natural numbers n
Correct Answer: B
Solution :
| [b] Let \[P(n):{{(1+x)}^{n}}\ge (1+nx)\] |
| For \[n=1,{{(1+x)}^{1}}=1+x\] |
| \[=1+1x\ge 1+1.x{{(1+x)}^{1}}\ge 1+1.x\] |
| For \[n=k,\] let \[P(k):{{(1+x)}^{k}}\ge (1+kx)\] is true. |
| For \[n=k+1,P(k+1):{{(1+x)}^{k+1}}\ge \{1+(k+1)x\}\] |
| is also true. |
| We will show \[P(k+1)\] is true. |
| Consider \[{{(1+x)}^{k+1}}={{(1+x)}^{k}}(1+x)\ge (1+kx)(1+x)\] \[[if\,\,x>-1]\] |
| \[=1+x+kx+k{{x}^{2}}\ge 1+x+kx\] \[[\because \,\,\,k>0\,\,and\,\,x>-1]\] |
| \[=1+(k+1)x\] |
| Thus, \[{{(1+x)}^{k+1}}\ge 1+(k+1)x,\,\,if\,\,x>-1\] |
You need to login to perform this action.
You will be redirected in
3 sec
