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Question

Find the middle term in the expansion of x3+9y10.


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Solution

Compute the middle term.

We know that the general form of a+bn is Tr+1=Crnan-rbr.

Here n=10,a=x3.b=9y.

Since n is even,

The middle term =n2+1th term.

=102+1th term

=6th term

Here T6=T5+1. That is,

r=5

So we can write that

T5+1=C510x310-59y5

=C510x3595y5=10!5!10-5!x35325y5=10!5!5!×x535×310×y5 ∴Crn=n!(n–r)!r!

=10×9×8×7×6×5!5×4×3×2×1×5!×x5×35×y5=10×9×8×7×6×355×4×3×2×1×x5×y5=61236x5y5

Hence, the middle term of the expansion is 61236x5y5.


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