What is the General Term of Binomial Expansion?

S3-SA1-0178

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The General Term of a Binomial Expansion is a formula that helps us find any specific term (like the 3rd term or the 7th term) in the expansion of a binomial expression like (a + b)^n without having to expand the whole thing. It's like having a shortcut to directly find a particular item in a long list.

Simple Example

Quick Example

Imagine you are making a rangoli design with many different colours. If someone asks you, 'What colour is the 5th flower from the left?', you don't need to count all the flowers from the beginning. If you had a rule that tells you the colour of the nth flower, you could directly find the 5th flower's colour. The General Term does this for binomial expansions.

Worked Example

Step-by-Step

Let's find the 3rd term in the expansion of (x + y)^4.

Step 1: Understand the formula for the general term: T(r+1) = C(n, r) * a^(n-r) * b^r. Here, 'n' is the power, 'a' is the first term, 'b' is the second term, and 'r' is one less than the term number you want.

---Step 2: Identify n, a, b, and the term number. For (x + y)^4, n = 4, a = x, b = y. We want the 3rd term, so r+1 = 3, which means r = 2.

---Step 3: Substitute these values into the formula: T(2+1) = C(4, 2) * x^(4-2) * y^2.

---Step 4: Calculate C(4, 2). C(n, r) = n! / (r! * (n-r)!). So, C(4, 2) = 4! / (2! * (4-2)!) = 4! / (2! * 2!) = (4 * 3 * 2 * 1) / ((2 * 1) * (2 * 1)) = 24 / 4 = 6.

---Step 5: Substitute the C(4,2) value back: T3 = 6 * x^2 * y^2.

---Answer: The 3rd term in the expansion of (x + y)^4 is 6x^2y^2.

Why It Matters

Understanding the General Term is super important in fields like Data Science and AI/ML, where complex patterns need to be simplified. Engineers use it to design structures and predict how systems behave. Even in Economics, it helps model growth and change, making it a foundational skill for many exciting future careers.

Common Mistakes

MISTAKE: Using 'r' as the term number directly in C(n, r) and the exponents. | CORRECTION: Remember that the formula is T(r+1), so if you want the 'k'th term, you use r = k-1. For example, for the 5th term, r = 4.

MISTAKE: Swapping 'a' and 'b' or their exponents. | CORRECTION: Always keep 'a' as the first term in the binomial and 'b' as the second. 'a' gets the (n-r) exponent, and 'b' gets the 'r' exponent.

MISTAKE: Forgetting to calculate the binomial coefficient C(n, r) correctly. | CORRECTION: Double-check your factorial calculations. C(n, r) = n! / (r! * (n-r)!) is crucial and often where small errors happen.

Practice Questions

Try It Yourself

QUESTION: What is the 4th term in the expansion of (a + b)^5? | ANSWER: 10a^2b^3

QUESTION: Find the 2nd term in the expansion of (2x + y)^3. | ANSWER: 12x^2y

QUESTION: If the 3rd term of an expansion (p + q)^n is 6p^2q^2, what is the value of n? | ANSWER: n = 4

MCQ

Quick Quiz

Which of the following is the correct general term for (x + y)^n?

C(n, r) * x^r * y^(n-r)

C(n, r) * x^(n-r) * y^r

C(n, r+1) * x^(n-r) * y^r

C(n, r) * x^(n-r) * y^(r+1)

The Correct Answer Is:

The general term formula is T(r+1) = C(n, r) * a^(n-r) * b^r. So for (x+y)^n, 'a' is x and 'b' is y, making option B correct. Options A swaps the exponents, and C and D use incorrect 'r' values or exponents.

Real World Connection

In the Real World

Imagine you're an engineer designing a new mobile phone antenna. The signal strength can be modeled using binomial expansions. The General Term helps you quickly calculate the strength at a specific point without needing to map out the entire signal pattern. This saves time and helps build better devices, like those used for 5G connectivity across India.

Key Vocabulary

Key Terms

BINOMIAL: An algebraic expression with two terms, like (a+b) | EXPANSION: Writing out all the terms of a binomial raised to a power | TERM: Each part of an algebraic expression separated by a plus or minus sign | COEFFICIENT: The number multiplied by a variable in an algebraic term, like the '6' in 6x^2y^2 | FACTORIAL: The product of an integer and all the integers below it, denoted by '!' (e.g., 4! = 4*3*2*1)

What's Next

What to Learn Next

Great job understanding the General Term! Next, you can explore how to find the middle term(s) of a binomial expansion, which uses this concept. You can also learn about specific cases like finding terms independent of 'x' or the greatest term, building on this strong foundation.