The overall term \[Tr+1\] in the binomial extension is given by \[Tr+1\text{ }=\text{ }nCr\text{ }an-r\text{ }br\]
Here the binomial is \[\left( 1+x \right)n\] with\[a\text{ }=\text{ }1\] , \[b\text{ }=\text{ }x\] and \[n\text{ }=\text{ }n\]
The \[\left( r+1 \right)th\] term is given by
\[T\left( r+1 \right)\text{ }=\text{ }nCr\text{ }1n-r\text{ }xr\]
\[T\left( r+1 \right)\text{ }=\text{ }nCr\text{ }xr\]
The coefficient of \[\left( r+1 \right)th\] term is \[nCr\]
The \[rth\] term is given by \[\left( r-1 \right)th\] term
\[T\left( r+1-1 \right)\text{ }=\text{ }nCr-1\text{ }xr-1\]
\[Tr\text{ }=\text{ }nCr-1\text{ }xr-1\]
∴ the coefficient of \[rth\] term is \[nCr-1\]
For \[\left( r-1 \right)th\] term we will take \[\left( r-2 \right)th\] term
\[Tr-2+1\text{ }=\text{ }nCr-2\text{ }xr-2\]
\[Tr-1\text{ }=\text{ }nCr-2\text{ }xr-2\]
∴ the coefficient of \[\left( r-1 \right)th\] term is \[nCr-2\]
Considering that the coefficient of\[\left( r-1 \right)th\] , \[rth\] and \[r+1th\] term are in proportion \[1:3:5\]
Accordingly,
By cross increase
\[\Rightarrow 5r\text{ }=\text{ }3n\text{ }\text{ }3r\text{ }+\text{ }3\]
\[\Rightarrow 8r\text{ }\text{ }3n\text{ }\text{ }3\text{ }=0\ldots \text{ }\ldots \text{ }\ldots \text{ }.2\]
We have \[1\] and \[2\] as
\[n\text{ }\text{ }4r\text{ }\pm \text{ }5\text{ }=0\ldots \text{ }\ldots \text{ }\ldots \text{ }1\]
\[8r\text{ }\text{ }3n\text{ }\text{ }3\text{ }=0\ldots \text{ }\ldots \text{ }\ldots \text{ }.2\]
Increasing condition \[1\] by number \[2\]
\[2n\text{ }-\text{ }8r\text{ }+10\text{ }=0\ldots \text{ }\ldots \text{ }\ldots \text{ }\ldots \text{ }.3\]
Adding condition \[2\] and \[3\]
\[2n\text{ }-\text{ }8r\text{ }+10\text{ }=0\]
\[-\text{ }3n\text{ }\text{ }8r\text{ }\text{ }3\text{ }=0\]
\[\Rightarrow -\text{ }n\text{ }=\text{ }-\text{ }7\]
\[n\text{ }=7\] and \[r\text{ }=\text{ }3\]