# Consider the binomial expansion of (1+7X)^23 in accending powers of x. Find the greatest numerical coefficient of the expansion and the terms of the expansion corresponding to it.

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`(1+7x)^23=sum_{r=0}^23 7^r C(23,r)x^r`

We have to find r such that

`7^(r-1)C(23,r-1)<7^rC(23,r)`

`and `

`7^rC(23,r)>7^(r+1)C(23,r+1)`

from these inequality

we have `20<r<21`

Thus greatest coefficient is

`C(23,21)xx7^21=23xx11xx7^21=C(23,20)xx7^20`

and term is

`T_{21}=C(23,20)(7x)^20`

`and`

`T_{22}=C(23,21)(7x)^(21)`