1. Find Middle term of `(3x-y)^8`

Solution:
`(3x-y)^8`

As `n=8` is even, so we have one middle term

`n/2+1=8/2+1=5^(th)` term

`"Here "a=3x,b=-y,n=8`

Now, `T_(r+1)=((n),(r))a^(n-r)b^r`

To find `T_5`, we let `r=4`

`:. T_5=T_(4+1)`

`=((8),(4))(3x)^4(-y)^4`

`=(8!)/(4!(8-4)!)(3x)^4(-y)^4`

`=(8*7*6*5)/(4*3*2*1)(81x^4)(y^4)`

`=70(81x^4)(y^4)`

`=5670x^4y^4`

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2. Find Middle term of `(3x-y)^9`

Solution:
`(3x-y)^9`

As `n=9` is odd, so we have two middle terms

`(n+1)/2=(9+1)/2=5^(th)` term and `(n+1)/2+1=(9+1)/2+1=6^(th)` term

`"Here "a=3x,b=-y,n=9`

Now, `T_(r+1)=((n),(r))a^(n-r)b^r`

To find `T_5`, we let `r=4`

`:. T_5=T_(4+1)`

`=((9),(4))(3x)^5(-y)^4`

`=(9!)/(4!(9-4)!)(3x)^5(-y)^4`

`=(9*8*7*6)/(4*3*2*1)(243x^5)(y^4)`

`=126(243x^5)(y^4)`

`=30618x^5y^4`

To find `T_6`, we let `r=5`

`:. T_6=T_(5+1)`

`=((9),(5))(3x)^4(-y)^5`

`=(9!)/(5!(9-5)!)(3x)^4(-y)^5`

`=(9*8*7*6)/(4*3*2*1)(81x^4)(-y^5)`

`=126(81x^4)(-y^5)`

`=-10206x^4y^5`

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3. Find Middle term of `(x/2+3y)^8`

Solution:
`(x/2+3y)^8`

As `n=8` is even, so we have one middle term

`n/2+1=8/2+1=5^(th)` term

`"Here "a=x/2,b=3y,n=8`

Now, `T_(r+1)=((n),(r))a^(n-r)b^r`

To find `T_5`, we let `r=4`

`:. T_5=T_(4+1)`

`=((8),(4))(x/2)^4(3y)^4`

`=(8!)/(4!(8-4)!)(x/2)^4(3y)^4`

`=(8*7*6*5)/(4*3*2*1)((x^4)/16)(81y^4)`

`=70((x^4)/16)(81y^4)`

`=(2835x^4y^4)/8`

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4. Find Middle term of `(x/2+3y)^9`

Solution:
`(x/2+3y)^9`

As `n=9` is odd, so we have two middle terms

`(n+1)/2=(9+1)/2=5^(th)` term and `(n+1)/2+1=(9+1)/2+1=6^(th)` term

`"Here "a=x/2,b=3y,n=9`

Now, `T_(r+1)=((n),(r))a^(n-r)b^r`

To find `T_5`, we let `r=4`

`:. T_5=T_(4+1)`

`=((9),(4))(x/2)^5(3y)^4`

`=(9!)/(4!(9-4)!)(x/2)^5(3y)^4`

`=(9*8*7*6)/(4*3*2*1)((x^5)/32)(81y^4)`

`=126((x^5)/32)(81y^4)`

`=(5103x^5y^4)/16`

To find `T_6`, we let `r=5`

`:. T_6=T_(5+1)`

`=((9),(5))(x/2)^4(3y)^5`

`=(9!)/(5!(9-5)!)(x/2)^4(3y)^5`

`=(9*8*7*6)/(4*3*2*1)((x^4)/16)(243y^5)`

`=126((x^4)/16)(243y^5)`

`=(15309x^4y^5)/8`

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