The ratio of the coefficient of  x^15 to the term independent of x in[X^2+(2/x)]^15 is what?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find what is the term independent of x in the binomial development of `[x^2+(2/x)]^15` .

`a_(k+1) = C_15^k*(x^2)^(15 - k)*(2/x)^k`

Since this term is independent of x, hence the power of x is zero.

`a_(k+1) = C_15^k*(x)^(30 - 2k)*(2^k)/(x^k)`

You need to equate like powers of x such that:

`0 = 30 - 2k -k =gt 30 - 3k = 0 =gt 30 = 3k =gt k = 10`

The term independent of x is `a_11` . The binomial coefficient of `a_11`  is `C_15^10` .

Apply the formula using factorials to find `C_15^10` .

`C_15^10 = (15!)/((10!)(5!)) =gt C_15^10 = (10!*11*12*13*14*15)/((10!)(5!))`

`C_15^10 = (11*12*13*14*15)/(6*2*2*5) = 11*13*7*3 = 3003`

You need to find the binomial coefficient of the term containing `x^15` .

`15 = 30 - 2k -k =gt 30 - 3k = 15 =gt 3k = 30 -15 =gt 3k = 15 =gt k = 5`

The term  containing `x^15`  is `a_6` . The binomial coefficient of `a_6`  is`C_15^5` .

`C_15^5 = (15!)/((5!)(10!)) =gt C_15^5 = 3003`

Hence the ratio of the term  term  containing x^15 to the term independent of x is:  term  containing `(C_15^10)/(C_15^5) = 3003/3003 = 1.`

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