find the coefficient of [tex] \sf {x}^{8} \: \: \: in the \\ expression \: 5{x}^{2}\:{2{x}^{5}+\frac{1}{x} }^{20} [\tex]
Answers
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
Answer:
2 ×16 ×16
=80
=3278