General and Middle Terms

General and Middle terms are used in binomial expansion which is a mathematical technique used to expand expressions of the form (a + b)n, where "a" and "b" are constants, and "n" is a positive integer. The expansion involves raising the binomial to a specific power and finding the individual terms that result from it. 

• In a binomial expansion, each term can be identified by its power of "a" and "b." 
• The power of "a" decreases with each subsequent term, while the power of "b" increases. 
• Among the terms in a binomial expansion, two types are often of interest: the general term and the middle term. 
• The general term represents any term in the expansion, allowing us to calculate specific terms by substituting the appropriate values. 
• On the other hand, the middle term refers to the term that appears in the middle of the expansion.

Read More: Binomial Expansion Formula

Key Terms: Binomial Theorem, Coefficients, General Term, Middle Term, Independent Terms, Odd, Even, Factorial, Combinations.

---

What is Binomial Theorem?

[Click Here for Sample Questions]

The Binomial Theorem provides a systematic way to expand expressions of the form (a + b)n, where "a" and "b" are constants and "n" is a positive integer. 

• It allows us to determine the coefficients and powers of the terms in the expansion without actually multiplying the binomial repeatedly.
• The Binomial Theorem states that for any positive integer "n," the expansion of (a + b)n can be expressed as:

(a + b)ⁿ = C(n, 0) * aⁿ * b⁰ + C(n, 1) * a^(n-1) * b¹ + C(n, 2) * a^(n-2) * b² + ... + C(n, r) * a^(n-r) * b^r + ... + C(n, n) * a⁰ * bⁿ

where C(n, r) represents the binomial coefficient, given by the formula:

C(n, r) = n! / (r! * (n-r)!)

Here, "!" denotes factorial.

• The binomial coefficient C(n, r) represents the number of ways to choose "r" items from a set of "n" items. 
• It determines the relative contribution of each term in the expansion.
• In each term, the power of "a" starts with "n" and decreases by one with each subsequent term, while the power of "b" starts with zero and increases by one.

Read More: Binomial Theorem for Positive Integral Indices

---

Binomial Coefficients

[Click Here for Sample Questions]

Binomial coefficients are mathematical values that appear as coefficients in the expansion of binomial expressions using the Binomial Theorem. 

• A binomial coefficient, denoted as C(n, r) or "n choose r," represents the number of ways to choose "r" objects from a set of "n" distinct objects without considering their order. It is defined by the formula:

C(n, r) = n! / (r! * (n - r)!)

• Binomial coefficients exhibit several notable properties. 
• They satisfy the symmetry property, where C(n, r) = C(n, n - r), reflecting the symmetry of choosing "r" objects out of "n." 
• Additionally, the sum of the binomial coefficients in a row of Pascal's triangle (a triangular array of numbers with each entry being the sum of the two numbers directly above it) is equal to 2 raised to the power of the row number.

1 → (x + y)⁰ = 1

1 1 → (x + y)¹ = x + y

1 2 1 → (x + y)² = x² + 2xy + y²

1 3 3 1 → (x + y)³ = x³ + 3x²y + 3xy² + y³

1 4 6 4 1 → (x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴

1 5 10 10 5 1 → (x + y)⁵ = x⁵ + 5x⁴y + 10x³y² + 10x²y³ + 5xy⁴ + y⁵

Example: Expand (2x - 3y)⁴.

Solution: Step 1: Determine the number of terms in the expansion.

Since the power is 4, there will be a total of (4 + 1) = 5 terms in the expansion.

Step 2: Identify the terms and their coefficients.

The terms in the expansion will have powers of (2x) and (-3y) that add up to 4. We can start from the highest power of (2x) and decrease it by one with each subsequent term.

Term 1: (2x)⁴

Using the formula for the binomial coefficient C(n, r), we have C(4, 0) = 1.

The first term is (1) * (2x)⁴ * (-3y)(4 - 0) = 16x⁴ * 81y⁴ = 1296x⁴y⁴.

Term 2: C(4, 1) = 4

The second term is (4) * (2x)³ * (-3y)(4 - 1) = 32x³ * (-27y) = – 864x³y.

Term 3: C(4, 2) = 6

The third term is (6) * (2x)² * (-3y)(4 - 2) = 6x² * 9y² = 54x²y².

Term 4: C(4, 3) = 4

The fourth term is (4) * (2x)¹ * (-3y)^{(4 - 3)} = 8x * (-3y) = -24xy.

Term 5: C(4, 4) = 1

The fifth term is (1) * (2x)⁰ * (-3y)^{(4 - 4)} = 1 * 1 * 1 = 1.

Step 3: Combine the terms to form the expanded expression.

The expanded form of (2x - 3y)⁴ is:

1296x⁴y⁴ - 864x³y + 54x²y² - 24xy + 1.

Read More: Combination Formula

---

General Term

[Click Here for Sample Questions]

The general term of a binomial expansion refers to an expression that represents any term in the expansion of a binomial raised to a specific power. It allows us to calculate the value of any term without explicitly writing out the entire expansion. 

• The general term can be determined using the binomial coefficient and the powers of the variables involved.
• In the expansion of (a + b)ⁿ, where "a" and "b" are constants and "n" is a positive integer, the general term is given by:

T(r+1) = C(n, r) * a^(n-r) * b^r

Here, T(r+1) represents the (r+1)-th term in the expansion, C(n, r) denotes the binomial coefficient, and a^(n-r) and br represent the powers of "a" and "b" respectively.

• In the general term formula, the power of "a" starts with (n-r) and decreases by one with each subsequent term, while the power of "b" starts with "r" and increases by one. 
• By substituting different values of "r" into the general term formula, we can calculate the value of specific terms in the binomial expansion.

• First Term = T(1) = C(n, 0) * a^(n-0) * b⁰ = C(n, 0) * aⁿ * 1 = C(n, 0) * aⁿ
• Second Term = T(2) = C(n, 1) * a^(n-1) * b¹
• Third Term = T(3) = C(n, 2) * a^(n-2) * b²
• nth Term = T(n) = C(n, n) * a^(n-n) * bⁿ = C(n, n) * bⁿ

Example: Find the general term of the expansion (3a - 2b)⁵.

Solution: Step 1: Finding out the number of terms in the expansion.

Since the power is 5, there will be a total of (5 + 1) = 6 terms in the expansion.

Step 2: Finding the general term using the binomial coefficient and the powers of "a" and "b".

The general term is given by the formula: T(r+1) = C(5, r) * (3a)^(5-r) * (-2b)^r.

Term 1: r = 0

Using the formula for the binomial coefficient C(5, 0) = 1.

The first term is (1) * (3a)^(5-0) * (-2b)⁰ = 35 * a⁵ * 1 = 243a⁵.

Term 2: r = 1

The second term is (5) * (3a)^(5-1) * (-2b)¹ = 5 * 34 * a⁴ * (-2b) = -1080a⁴b.

Term 3: r = 2

The third term is (10) * (3a)^(5-2) * (-2b)² = 10 * 33 * a³ * (2b)² = 2160a³b².

Term 4: r = 3

The fourth term is (10) * (3a)^(5-3) * (-2b)³ = 10 * 32 * a² * (-2b)³ = -2160a²b³.

Term 5: r = 4

The fifth term is (5) * (3a)^(5-4) * (-2b)⁴ = 5 * 31 * a¹ * (2b)⁴ = 720ab⁴.

Term 6: r = 5

The sixth term is (1) * (3a)^(5-5) * (-2b)⁵ = 1 * 30 * a⁰ * (-2b)⁵ = -32b⁵.

Step 3: Combining the terms to form the expanded expression.

The expanded form of (3a - 2b)⁵, using the general terms we found, is

243a⁵ - 1080a⁴b + 2160a³b² - 2160a²b³ + 720ab⁴ - 32b⁵.

Read More: Poisson Distribution Formula

---

Middle Term

[Click Here for Sample Questions]

In a binomial expansion, the middle term refers to the term that appears at the center of the expansion. The position of the middle term depends on whether the power "n" of the binomial is odd or even. 

Odd Power (n is odd):

When the power "n" is odd, the middle term occurs at the center of the expansion, with an equal number of terms on either side. The middle term is the term that appears at position (n+1)/2.

• The equation for the middle term is

Middle Term = C(n, (n+1)/2) * a^((n+1)/2) * b^((n+1)/2)

• Here, C(n, (n+1)/2) represents the binomial coefficient associated with the middle term, calculated using the formula:

C(n, (n+1)/2) = n! / ((n+1)/2)! * ((n-1)/2)!

• The powers of "a" and "b" in the middle term are both ((n+1)/2).

Even Power (n is even):

When the power "n" is even, there are two middle terms in the expansion, located symmetrically around the center. The middle terms are at positions n/2 and (n/2) + 1.

• The equations for the middle terms are:

Middle Term 1 = C(n, n/2) * a^(n/2) * b^(n/2)

Middle Term 2 = C(n, (n/2) + 1) * a^{((n/2) + 1)} * b^((n/2))

• For both middle terms, C(n, n/2) and C(n, (n/2) + 1) represent the corresponding binomial coefficients.

• In the first middle term, the powers of "a" and "b" are both n/2.
• In the second middle term, the powers of "a" and "b" are ((n/2) + 1) and (n/2), respectively.
• The first middle term occurs at position n/2, and the second middle term occurs at position (n/2) + 1.

Example (Odd Case): Find the middle term of expansion (2x - 3y)⁵.

Solution: Step 1: Determine the number of terms in the expansion.

Since the power is 5, there will be a total of (5 + 1) = 6 terms in the expansion.

Step 2: Find the middle term.

Since the power is odd, the middle term will be the term at position (n+1)/2 = (5+1)/2 = 3.

The third term is the middle term: C(5, 3) * (2x)² * (-3y)³ = 10 * 4x² * (-27y³) = -1080x²y³.

Therefore, the middle term of the expansion (2x - 3y)⁵ is -1080x²y³.

Example (Even Case): Find the middle term of expansion (2x - 3y)⁶.

Solution: Step 1: Determine the number of terms in the expansion.

Since the power is 6, there will be a total of (6 + 1) = 7 terms in the expansion.

Step 2: Find the middle terms.

Since the power is even, there will be two middle terms at positions n/2 = 6/2 = 3 and (n/2) + 1 = 3 + 1 = 4.

The third term is the first middle term: C(6, 3) * (2x)³ * (-3y)³ = 20 * 8x³ * (-27y³) = – 4320x³y³.

The fourth term is the second middle term: C(6, 4) * (2x)⁴ * (-3y)² = 15 * 16x⁴ * 9y² = 2160x⁴y².

Therefore, the middle terms of the expansion (2x - 3y)⁶ are – 4320x³y³ and 2160x⁴y².

---

Independent Term of the Binomial Expansion

[Click Here for Sample Questions]

The independent term of a binomial expansion refers to the constant term that does not contain any variables or variables raised to a power. It is also known as the term with zero degrees.

To determine the independent term in a binomial expansion, we need to find the term where all the variables have an exponent of zero.

Example: Find the independent term of the expansion (2x - 3)⁴.

Solution: Step 1: Determine the number of terms in the expansion.

Since the power is 4, there will be a total of (4 + 1) = 5 terms in the expansion.

Step 2: Find the independent term.

The independent term is the term where all variables have an exponent of zero.

In the expansion of (2x - 3)⁴, the independent term will be when the variable "x" is raised to the power of zero, and the constant term "-3" is raised to the power of (4 - 0) = 4.

The independent term is: C(4, 0) * (2x)⁰ * (-3)⁴ = 1 * 1 * 81 = 81.

Therefore, the independent term of the binomial expansion (2x - 3)⁴ is 81.

Read More: Pascal's Law and Its Applications

---

Things to Remember

• The general term of a binomial expansion is represented by the formula: T(r+1) = C(n, r) * a(n-r) * br.
• The binomial coefficient C(n, r) calculates the number of ways to choose "r" elements from a set of "n" elements and is given by the formula: C(n, r) = n! / (r! * (n-r)!).
• The middle term(s) in a binomial expansion depends on whether the power "n" is odd or even.
• When "n" is odd, there is a single middle term located at position (n+1)/2 in the expansion.
• When "n" is even, there are two middle terms located at positions n/2 and (n/2)+1 in the expansion.
• The middle term(s) exhibit symmetry in the expansion, with an equal number of terms on either side.