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Algebra of Complex Numbers

If 1 + x1...

Question

If $(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . C_{15} x^{15},$
find the value of $C_{2} + 2 C_{3} + 3 C_{4} + . . . . . 14 C_{15}$

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Solution

$(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . C_{15} x^{15}$
$∴ \frac{(1 + x)^{15}}{x} = \frac{C_{0}}{x} + C_{1} + C_{2} x + C_{3} x^{2} . . . . C_{15} x^{14}$
Differentiate both sides w.r.t. x
$\frac{x \cdot 15 (1 + x)^{14} - 1 \cdot (1 + x)^{15}}{x^{2}}$
$= - \frac{C_{0}}{x^{2}} + C_{2} + 2 C_{3} x + 3 C_{4} x^{2} . . . .14 C_{15} x^{13}$
Putting x = 1 on both sides
${15.2}^{14} - 2^{15} = - C_{0} + C_{2} + 2 C_{3} + 3 C_{4} . . . .14 C_{15}$
$2^{14} (15 - 2) + 1 = C_{2} + 2 C_{3} + 3 C_{4} . . . .14 C_{15}$
The given series = 2 $^{14}$ .13 + 1 = 219923.

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Similar questions

(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + \dots \dots + C_{15} x^{15},

find the value of

C_{2} + 2 C_{3} + 3 C_{4} + \dots \dots + 14 C_{15}

(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . + C_{15} x^{15},

C_{2} + 2 C_{3} + 3 C_{4} + . . . . . . + 14 C_{15} =

If $(1 + x)^{15}$ = $C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . . + C_{15} x^{15}$ , then

$C_{2} + 2 C_{3} + 3 C_{4} + . . . . . . . . + 14 C_{15}$ =

{(1 + x)}^{15} = C_{0} + C_{1} . x + C_{2} . x^{2} + . . . . + C_{15} . x^{15}

C_{2} + 2 C_{3} + 3 C_{4} + . . . . + 14 C_{15} = a 2^{b} + c,

a + b + c

f (x) = (1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{15} x^{15},

f (2) =

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