If fx-1-2x-3x2-4x3-17x16-18x17- then the coefficient of x15 in fx-1 is-are

F(x)=1 2x+3x^2 4x^3+⋯+17x^16 18x^17 xf(x) = x 2x^2+3x^3+⋯ 16x^16+17x^17 18x^18 Adding both, we get ⇒ (1+x)f(x)= 1 x+x^2 x^3+⋯+x^16 x^17 18x^18 ⇒ nbsp; ...

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Standard XII

Mathematics

Square Root of a Complex Number

If fx=1-2x+3x...

Question

If $f (x) = 1 - 2 x + 3 x^{2} - 4 x^{3} + \dots + 17 x^{16} - 18 x^{17}$ , then the coefficient of $x^{15}$ in $f (x - 1)$ is/are

16 \cdot^{18} C_{16} - 17 \cdot^{18} C_{17}

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^{18} C_{17} - 18^{18} C_{16}

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^{19} C_{17} - 19^{18} C_{16}

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17 \cdot^{18} C_{16} - 16 \cdot^{19} C_{16}

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Solution

The correct option is C $^{19} C_{17} - 19^{18} C_{16}$
$f (x) = 1 - 2 x + 3 x^{2} - 4 x^{3} + \dots + 17 x^{16} - 18 x^{17} x f (x) = x - 2 x^{2} + 3 x^{3} + \dots - 16 x^{16} + 17 x^{17} - 18 x^{18}$
Adding both, we get
$\Rightarrow (1 + x) f (x) = 1 - x + x^{2} - x^{3} + \dots + x^{16} - x^{17} - 18 x^{18} \Rightarrow (1 + x) f (x) = \frac{(- x)^{18} - 1}{- x - 1} - 18 x^{18} \Rightarrow f (x) = \frac{1 - (x)^{18}}{(x + 1)^{2}} - 18 \frac{x^{18}}{(1 + x)}$
Replacing $x \to x - 1$ , we get
$\Rightarrow f (x - 1) = \frac{1 - (x - 1)^{18}}{x^{2}} - 18 \frac{(x - 1)^{18}}{x}$
Coefficient of $x^{15}$
$= -^{18} C_{1} (- 1)^{1} - 18^{18} C_{2} (- 1)^{2} =^{18} C_{17} - 18^{18} C_{16} =^{19} C_{17} -^{18} C_{16} - 18^{18} C_{16} =^{19} C_{17} - 19^{18} C_{16}$

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Square Root of a Complex Number

Standard XII Mathematics

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If f(x)=1−2x+3x2−4x3+⋯+17x16−18x17, then the coefficient of x15 in f(x−1) is/are

If $f (x) = 1 - 2 x + 3 x^{2} - 4 x^{3} + \dots + 17 x^{16} - 18 x^{17}$ , then the coefficient of $x^{15}$ in $f (x - 1)$ is/are