find third term in the expansion of (2x + 5/√x)^5
Answers
Answer
All binomials to powers can be written as
(
a
+
b
)
x
While expanded, two rules come forward.
The result can be written as the addition of all powers of a times all powers of b, where the power on the a term plus the power on the b term is equal to x.
For example,
(
a
+
b
)
2
can be written as
a
2
+
2
a
b
+
b
2
. On the
a
2
term, the power a is being raised to is 2, and the power that b is being raised to is 0.
b
0
is equal to 1, and
1
⋅
a
2
=
a
2
, so the term is
a
2
. This will work for the other terms as well
2.There are coefficients on the terms, that increase as you reach the middle. For the sake of a (fairly) short answer, I'll assume you know Pascal's triangle.
If you know Pascal's triangle, just assign each row as being the coefficients when (a+b) is raised to the power of x-1. For example, the 4th row of Pascal's Triangle produces the result of 1 3 3 1, so if you raise (a+b) to the 4-1, or third power (equivalent to
(
a
+
b
)
3
), you get the coefficients of 1 3 3 1. You then assign
a
3
to the first coefficient,
a
2
b
to the second,
a
b
2
to the third, and
b
3
to the fourth, and then add them up, ending up with
a
3
+
3
a
2
b
+
3
b
2
a
+
b
3
.
Now, for your problem input 2x as a, 5 as b, and 5 as x. We are only looking for the third term, so we go to the value where a is only being raised to the 3rd power. As a result, b must be raised to the second power, so the powers on a and b are equal to 5. This produces the result of
(
2
x
)
3
⋅
5
2
=
8
x
3
⋅
25
=
200
x
3
. Finally, using Pascal's Triangle, we know the third term is 10, so we multiply that by the amount already there, obtaining the result of
10
⋅
200
x
3
=
2000
x
3
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Correct option is
D
41
(1+xlog2x)5
T3=5C2⋅(xlog2x)2=2560
⇒10⋅x2log2x=2560
⇒x2log2x=256
⇒2(log2x)2=log2