This is the worked solution for
the fun question.
Question
Using Binomial
Theorem, obtain the first 3 terms in the expansion
(1-x)/[square
root(4-x)]
Show that if x =
2/5, then (1-x)/[square root(4-x)] = - 1/(square root
10).
Hence, calculate
the value of - 1/(square root 10) correct to 3 significant
figures.
Answer
Step 1 :
Use Binomial Expansion to
obtain the first three terms of the expression
(1-x)/[square
root(4-x)]
= (1-x)/[square
root(4(1-x/4))]
=
(1/2)(1-x)/[square root(1-x/4)]
=
(1/2)(1-x)(1-x/4)^(-1/2)
=
(1/2)(1-x)[1 +
(-1/2)(-x/4) + {(-1/2)(-3/2)}/{(1x2)}(-x/4)^2 + .......]
Use Binomial Expansion
= (1/2)(1-x)[1 +
x/8 + (3/128)x^2 + .......}
= (1/2)[1+ x/8 +
(3/128)x^2 - x - 1/8x^2 + ....... ]
= (1/2)[1 -
(7/8)x - (13/128)x^2 + ........]
Step 2 : To show
that if x = 2/5, then
(1-x)/[square
root(4-x)] = 1/(square root 10)
Substitute x =
2/5 into
(1-x)/[square
root(4-x)]
= (1 -
2/5)/[square root (4 - 2/5)]
= (3/5)/[square
root (18/5)]
= [Square
root(9/25] / [square root (18/5)]
Make 3/5 into [Square
root(9/25)]
= square root
[(9/25)/(18/5)]
= square root
[1/10]
= 1/ square root
10 (shown)
Step 3 : Use of
the first three terms in the expansion to calculate the value
of
1/(square root 10) correct to 3 significant
figures
Substitute x =
2/5 into the first three terms in the expansion
1/square root 10
= (1/2)[1 - (7/8)x - (13/128)x^2 + ........]
= (1/2)[1 - (7/8)(2/5) - (13/128)(2/5)^2 + ........]
= 0.317 (correct to 3 significant figures)
Thank you for your kind attention.
Regards,
ahm97sic