Determine the fifth term in the expansion of : (2x - 1/2x)^8? how do I determine the 5th term?

Let's start with various powers of a + b
(a + b)^0 = 1
(a + b)^1 = a + b
(a + b)² = a² + 2ab + b²
(a + b)^3 = a^3 + 3a²b + 3ab² + b^3
(a + b)^4 = a^4 + 4a^3b + 6a²b² + 4ab^3 + b^4

I'm not sure if you see the pattern, but focus on the number of terms and the exponents. When raising to the nth power, you have n+1 terms.
If n = 0, you have 1 term
If n = 1, you have 2 terms
if n = 2, you have 3 terms
etc.

Now look at the sequence of terms and their exponents. Apart from n=0, you always have terms involving a and b. The exponents always add up to n.

For for example, n = 4, the sequence of terms is a^4, a^3b, a^2b^2, ab^3, b^4

Notice how the powers of a go down 4, 3, 2, 1, 0. And the powers of b go up 0, 1, 2, 3, 4. In all cases, the sum of the two exponents is 4.

So the only remaining part is to figure out the coefficients (the numbers in front of each term). If you are familiar with Pascal's Triangle, it's just the terms in the nth row

Row 0 -------------> 1
Row 1 ----------> 1 .... 1
Row 2 -------> 1 .... 2 .... 1
Row 3 ----> 1 .... 3 .... 3 .... 1
Row 4 --> 1 ... 4 .... 6 .... 4 .... 1

Okay, so let's figure out (a + b)^8:

The 9 terms (8+1) will be a^8, a^7b, a^6b^2, a^5b^3, a^4b^4, a^3b^5, a^2b^6, ab^7, b^8

And the coefficients will be the 8th row of Pascal's triangle:
C(8,0) = 1
C(8,1) = 8
C(8,2) = 28
C(8,3) = 56
C(8,4) = 70
C(8,5) = 56
C(8,6) = 28
C(8,7) = 8
C(8,8) = 1


So the full sequence is:
a^8 + 8a^7b + 28a^6b^2 + 56a^5b^3 + 70a^4b^4 + 56a^3b^5 + 28a^2b^6 + 8ab^7 + b^8

Now I'm not sure if your teacher starts counting from the first term or the zeroth term:

If the first term is a^8, then the fifth term is 70a^4b^4.
If the zeroth term is a^8, then the fifth term is 56a^3b^5.

Just plug in a = 2x and b = 1/(2x) and you'll have the 5th term.

(x + y)^n = C(n,0)(x^n) - C(n,1)[x^(n-1)]y + C(n,2)[x^(n-2)](y^2) - …….
The rth term = (-1)^(r-1) C(n,r-1) [x^(n-r+1)] [y^(r-1)]

In expansion of [2x - (1/2x)]^8, the 5th term
= (-1)^(5-1) C(8,5-1) [(2x)^(8-5+1)] [(1/2x)^(5-1)]
= (-1)^4 C(8,4) (2x)^4 (1/2x)^4
= 8!/(4!4!)
= 70

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