Binominal Theorm (1 Viewer)

ayehann · May 31, 2009

Hey guys, i get a different answer to the back of the textbook, so just want to make sure that im not worng... what answers do you guys get for this.?

Find the value of k if the coefficients of x^{k+1} and x^{k} in (3x + 2)^{19} are equal. Note that the question asks for X6K and X^k+1 not X^k-1 and X^k

ayehann · May 31, 2009

btw this is question 8 from SB Jones 3 Unit Book 2. if you have that book. tkae a look at it in there.

Drongoski · May 31, 2009

ayehann said:
btw this is question 8 from SB Jones 3 Unit Book 2. if you have that book. tkae a look at it in there.

Is k = 11 ?

Drongoski · May 31, 2009

Yes k = 11.

$\100dpi (3x+2)^{19} = (2+3x)^{19}= \ \sum_{i=0}^{19}\binom{19}{i}2^{19-i}(3x)^{i}\\ \\ \therefore\ \ coeffs\ of\ x^{k}\ and\ x^{k+1}\ equal\ \ \Rightarrow \binom{19}{k}2^{19-k}3^{k}\ =\ \binom{19}{k+1}2^{19-(k+1)}\ 3^{k+1}\\ \\ \therefore\ \ \frac{19!}{k!(19-k)!}\ 2^{19-k}3^{k}\ =\ \frac{19!}{(19-k-1)!(k+1)!}2^{19-(k+1)}3^{k+1}\\ \\\ \ \ \ \ =\ \ \ \frac{19!(19-k)}{(19-k)!\ k!(k+1)}\cdot \frac{3}{2}2^{19-k}3^{k}\\ \\ \ \therefore\ \ \frac{3(19-k)}{2(k+1)}\ =\ 1\ \Rightarrow \ k=11$

Binominal Theorm (1 Viewer)

ayehann

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ayehann

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Drongoski

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Drongoski

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