Math, asked by Anonymous, 11 months ago

find the coefficient of [tex] \sf {x}^{8} \: \: \: in the \\ expression \: 5{x}^{2}\:{2{x}^{5}+\frac{1}{x} }^{20} [\tex]​

Answers

Answered by rudrakshkumar16
2

Step-by-step explanation:

(1−2x)

18

+ax(1−2x)

18

+bx

2

(1−2x)

18

Coefficient of

x^{3} x

3

: (-2)^{3} \: ^{18}C_3(−2)

3

18

C

3

+ $$a\times(-2)^{2} \times^{18}C_2

++b\times(-2) \times{^{18}C_1} = 0 $$

$$ \displaystyle \frac{4\times (17\times 16)}{(3\times 2)}-2a\cdot

\displaystyle \frac{17}{2}+b=0 \cdots (i)$$

Coefficient of

x^{4} x

4

: (-2)^{4} \: ^{18}C_4(−2)

4

18

C

4

+ $$a\times(-2)^{3} \times^{18}C_3

++b\times(-2)^{2} \times{^{18}C_2} = 0 $$

(4\times 20)-2a\cdot \displaystyle \frac{16}{3}+b=0 \cdots (ii) (4×20)−2a⋅

3

16

+b=0 ⋯(ii)

From equation (i)(i) and (ii)(ii), we get

$$ 4\left

( \displaystyle \frac{17\times 8}{3} -20\right )+2a \left(

\displaystyle \frac{16}{3} -\displaystyle \frac{17}{2}\right )=0$$

4\left ( \displaystyle \frac{17\times 8-60}{3} \right )+ \displaystyle \frac{2a(-19)}{6}=04(

3

17×8−60

)+

6

2a(−19)

=0

a=\displaystyle \frac{4\times 76\times 6}{3\times 2\times 19}a=

3×2×19

4×76×6

\Rightarrow a=16 ⇒a=16

\Rightarrow b=\displaystyle \frac{2\times 16\times 16}{3}-80=\displaystyle \frac{272}{3}⇒b=

3

2×16×16

−80=

3

272

Answered by Anonymous
0

Answer:

2 ×16 ×16

=80

=3278

Hope this helps

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