Find the coefficient of x24 in the expansion of -x2 - 3a-x-15-

nbsp;General term of the expansion (a + x)n is , nbsp; Tr+1 = nCr an r xr = ((n(n 1)(n 2)....(n r+1))/r!) an r xr. The general term ((r + 1)th term) is ...

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Standard XII

Mathematics

Integration of Irrational Algebraic Fractions - 1

Find the coef...

Question

Find the coefficient of x²⁴ in the expansion of (x² - 3a/x)¹⁵.

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Solution

General term of the expansion (a + x)ⁿ is ,
T_r₊₁ = ⁿC_ra^n-r x^r = ((n(n-1)(n-2)....(n-r+1))/r!) a^n-r x^r.
The general term ((r + 1)th term) is
(x² - 3a/x)¹⁵ = ¹⁵C_r(x²)^15-r (-3a/x)^r =(-1)^r ¹⁵C_r x^30-2r (3^ra^r/x^r) =(-1)^r ¹⁵C_r 3^r a^r x^30-3r
If this term contains x²⁴, then 30-3r = 24
3r = 6
r = 2.
Therefore, the coefficient of x²⁴ = ¹⁵C₂ 9a²=945a²

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Find the coefficient of x24 in the expansion of (x2 - 3a/x)15.

Find the coefficient of x²⁴ in the expansion of (x² - 3a/x)¹⁵.