General Term and Middle Term of Binomial Expansion
The binomial theorem provides a formula for expanding expressions of the form (a + b)n, where n is a non-negative integer. The expansion is given by:
(a + b)n = C(n, 0) an b0 + C(n, 1) an-1 b1 + C(n, 2) an-2 b2 + ... + C(n, r) an-r br + ... + C(n, n) a0 bn
Here, C(n, r), also written as nCr or (nr), represents the binomial coefficient, calculated as n! / (r! * (n-r)!).
The General Term
In the binomial expansion of (a + b)n, each term follows a specific pattern. The term that contains br is the (r+1)-th term. This is often referred to as the general term because by varying the value of r from 0 to n, we can obtain any term in the expansion.
The general term, denoted as Tr+1, is given by the formula:
Tr+1 = C(n, r) an-r br
In this formula:
- n is the exponent of the binomial.
- r is the power of the second term (b).
- C(n, r) is the binomial coefficient for that term.
- an-r is the power of the first term (a).
It's crucial to remember that r starts from 0 for the first term (T1). So, if you are asked for the 5th term, you would set r+1 = 5, which means r = 4.
Example 1: Finding a Specific Term
Find the 4th term in the expansion of (x + 2y)7.
Here, a = x, b = 2y, and n = 7.
We need the 4th term, so r + 1 = 4, which implies r = 3.
Using the general term formula: Tr+1 = C(n, r) an-r br T4 = C(7, 3) x7-3 (2y)3
Calculate the binomial coefficient C(7, 3): C(7, 3) = 7! / (3! * (7-3)!) = 7! / (3! * 4!) = (7 * 6 * 5 * 4!) / ((3 * 2 * 1) * 4!) = (7 * 6 * 5) / 6 = 35
Now substitute back into the term: T4 = 35 * x4 * (23 y3) T4 = 35 * x4 * (8 y3) T4 = 280 x4 y3
So, the 4th term in the expansion of (x + 2y)7 is 280x4y3.
Example 2: Finding the Term Independent of x
Find the term independent of x in the expansion of (x - 1/x)6.
Here, a = x, b = -1/x, and n = 6.
The general term is: Tr+1 = C(6, r) x6-r (-1/x)r Tr+1 = C(6, r) x6-r (-1)r (x-r) Tr+1 = C(6, r) (-1)r x6-r-r Tr+1 = C(6, r) (-1)r x6-2r
For the term to be independent of x, the power of x must be 0. So, we set the exponent equal to zero: 6 - 2r = 0 2r = 6 r = 3
Now, we find the term by substituting r = 3 back into the general term formula: T3+1 = T4 = C(6, 3) (-1)3 x6-2(3) T4 = C(6, 3) * (-1) * x0
Calculate C(6, 3): C(6, 3) = 6! / (3! * 3!) = (6 * 5 * 4 * 3!) / ((3 * 2 * 1) * 3!) = (6 * 5 * 4) / 6 = 20
So, the term is: T4 = 20 * (-1) * 1 = -20
The term independent of x is -20.
Middle Term(s)
The middle term(s) in a binomial expansion depend on whether the exponent n is even or odd.
Case 1: When n is Even
If n is an even number, there is exactly one middle term. The position of this middle term is (n/2) + 1.
To find the middle term, we set r + 1 = (n/2) + 1. This means r = n/2.
The middle term is then: T(n/2) + 1 = C(n, n/2) an - n/2 bn/2 T(n/2) + 1 = C(n, n/2) an/2 bn/2
Example 3: Middle Term when n is Even
Find the middle term in the expansion of (2x - 3)8.
Here, n = 8, which is even.
The position of the middle term is (8/2) + 1 = 4 + 1 = 5.
So, we need to find the 5th term, which means r + 1 = 5, or r = 4.
Using the general term formula with a = 2x, b = -3, n = 8, and r = 4: T5 = C(8, 4) (2x)8-4 (-3)4 T5 = C(8, 4) (2x)4 (-3)4
Calculate C(8, 4): C(8, 4) = 8! / (4! * 4!) = (8 * 7 * 6 * 5 * 4!) / ((4 * 3 * 2 * 1) * 4!) = (8 * 7 * 6 * 5) / 24 = 70
Now substitute back: T5 = 70 * (16x4) * (81) T5 = 70 * 16 * 81 * x4 T5 = 1120 * 81 * x4 T5 = 90720 x4
The middle term is 90720x4.
Case 2: When n is Odd
If n is an odd number, there are two middle terms. Their positions are (n+1)/2 and (n+1)/2 + 1.
To find the first middle term, we set r + 1 = (n+1)/2. This means r = (n+1)/2 - 1 = (n-1)/2. The first middle term is: T(n-1)/2 + 1 = C(n, (n-1)/2) an - (n-1)/2 b(n-1)/2 T(n+1)/2 = C(n, (n-1)/2) a(n+1)/2 b(n-1)/2
To find the second middle term, we set r + 1 = (n+1)/2 + 1. This means r = (n+1)/2. The second middle term is: T(n+1)/2 + 1 = C(n, (n+1)/2) an - (n+1)/2 b(n+1)/2 T(n+3)/2 = C(n, (n+1)/2) a(n-1)/2 b(n+1)/2
Example 4: Middle Terms when n is Odd
Find the middle terms in the expansion of (p + q)9.
Here, n = 9, which is odd.
The positions of the middle terms are: (9+1)/2 = 10/2 = 5 and (9+1)/2 + 1 = 5 + 1 = 6.
So, we need to find the 5th term and the 6th term.
For the 5th term: r + 1 = 5, so r = 4. T5 = C(9, 4) p9-4 q4 T5 = C(9, 4) p5 q4
Calculate C(9, 4): C(9, 4) = 9! / (4! * 5!) = (9 * 8 * 7 * 6 * 5!) / ((4 * 3 * 2 * 1) * 5!) = (9 * 8 * 7 * 6) / 24 = 126
So, the 5th term is: T5 = 126 p5 q4
For the 6th term: r + 1 = 6, so r = 5. T6 = C(9, 5) p9-5 q5 T6 = C(9, 5) p4 q5
Calculate C(9, 5): C(9, 5) = 9! / (5! * 4!) = C(9, 4) = 126 (using the property C(n, r) = C(n, n-r))
So, the 6th term is: T6 = 126 p4 q5
The middle terms are 126p5q4 and 126p4q5.
Special Cases for General Term
When dealing with binomial expansions of the form (a + bx)n or (a - bx)n, the general term takes a slightly modified form.
For (a + bx)n: Tr+1 = C(n, r) an-r (bx)r Tr+1 = C(n, r) an-r br xr
For (a - bx)n: Tr+1 = C(n, r) an-r (-bx)r Tr+1 = C(n, r) an-r (-1)r br xr
The sign of the term depends on whether r is even or odd when there's a minus sign in the binomial.
Example 5: General Term with Negative Sign
Find the 3rd term in the expansion of (3 - 2x)5.
Here, a = 3, b = -2x, and n = 5.
We need the 3rd term, so r + 1 = 3, which means r = 2.
Using the general term formula: Tr+1 = C(n, r) an-r br T3 = C(5, 2) (3)5-2 (-2x)2 T3 = C(5, 2) (3)3 (-2)2 x2
Calculate C(5, 2): C(5, 2) = 5! / (2! * 3!) = (5 * 4 * 3!) / ((2 * 1) * 3!) = (5 * 4) / 2 = 10
Substitute back: T3 = 10 * (27) * (4) * x2 T3 = 10 * 108 * x2 T3 = 1080 x2
The 3rd term is 1080x2.
Applications in Problem Solving
Understanding the general term is fundamental to solving various problems related to binomial expansions. These include:
- Finding a specific term (e.g., 5th term, 8th term).
- Finding the term independent of x (i.e., the constant term).
- Finding the coefficient of a specific term (e.g., the coefficient of x7).
- Finding the middle term(s).
- Problems involving ratios of terms.
The key is always to set up the general term correctly based on the given binomial expression and then use the conditions of the problem to solve for r or other unknowns.