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If x is a positive real number different from 1, then prove that the numbers `1/(1+sqrtx),1/(1-x),1/(1-sqrtx),…` are in A.P. Also find their common difference.

Text Solution

Verified by Experts

The correct Answer is:
Common diff. `= (sqrt(x))/(1-x)`
We have
`1/(1-x)-1/(1+sqrtx)=1/((1-sqrtx)(1+sqrtx))-1/(1+sqrtx)=sqrtx/((1-x))`
and `1/(1-sqrtx)-1/(1-x)=(1+sqrtx-1)/(1-x)=sqrtx/(1-x)`
Therefore, the given numbers are in A.P. with common difference `sqrtx//(1-x)`.
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