InterviewSolution
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InterviewSolution
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Find the coefficent of x^(32) and 1/x^(17) " in the expansion of "(x^(4)-1/x^(3))^(15) |
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Answer» Solution :The general term in the GIVEN expansion is given by `T_(r+1) = (-1)^(r) xx .^(15) C _(r) xx (x^(4))^((15-r)) xx (1/(x^(3)))^(r)` `rArr T _(r+1) = (-1)^(r) xx .^(15) C _(r) xx x ^((60-7r)).""` … (i) Putting `60- 7r = 32, " we get " 7r = 28 rArr r = 4 rArr r+1 = 5.` Now, `T_(5)= T_(4+1) = (-1)^(4) xx .^(15) C _ (4) xx x^((60-28)) = .^(15 ) C _(4) xx x^(32)`. `:. " coefficent of" x^(32) = .^(15) C _(4) ((15 xx 14 xx 13 xx 12)/(4 xx 3 xx 2 xx 1 ))= 1365.` Let `T_ (s+1) " be the term containing "x^(-17) ""[:' 1/(x^(17)) = x^(-17)].` Then,` T_(s+1) = (-1) ^(s) xx .^(15) C _(s) xx x^((60-7s))""` ...(ii) [putting r = s in(i)] Putting `60-7s = -17, " we get " s = 11 and " therefore " , s+ 1 = 12.` Thus, the 12th term contains `x^(-17)`. Now, `T_(12) = T _((11+1))` ` = (-1)^(11) xx .^15 C _(11) xx x^((60-77)""` [putting s = 11 in (ii)] `=.-^15C_(4) xx x^(-17).` `:. " coefficent of " x^(-17) =-.^(15)C_(4)= - ((15 xx 14 xx 13 xx 12)/(4 xx 3 xx 2 xx 1 ))=-1365.` So,the coefficent of `x^(-17)`in the given expansion is-1365. HENCE, the coefficent of `x^(32)" is 1365 and that of " x^(-17) is -1365.` |
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