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General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced PDF Download

Binomial Theorem Formula – General Term

General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
By the Binomial theorem formula, we know that there are (n + 1) terms in the expansion of (a+b)n. Now, let’s say that T1, T2, T3, T4, … Tn+1 are the first, second, third, fourth, … (n + 1)th terms, respectively in the expansion of (a+b)n. Therefore,
General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
Generalizing it, we have the formula for the general term:
General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
where 0 ≤ r ≤ n. Let’s look at an example now.

Example 1
Find the fourth term in the expansion of (3x–y)7.
Solution. In this example, a = 3x, b = – y, and n = 7. From the above formula, we have
General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
To find the fourth term, T4, r = 3. Therefore,
General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
Hence, the fourth term in the expansion of (3x–y)7 = – 2835x4y3

Binomial Theorem Formula – Middle Term

When you are trying to expand (a+b)n and ‘n’ is an even number, then (n + 1) will be an odd number. Which means that the exapnsion will have odd number of terms. In this case, the middle term will be the (n/2 + 1)th term.

For example, if you are expanding (x+y)2, then the middle term will be the (2/2 + 1) = 2nd term. (n/2 + 1)th term is also denoted as (n+2/2)th term.

When you are trying to expand (a+b)n and ‘n’ is an odd number, then (n + 1) will be an even number. Hence, there are two middle terms: (n+1/2)th term and (n+3/2)th term.

For example, if you are expanding (x+y)3, then the middle terms will be, (3+1/2) = 2nd term and (3+3/2) = 3rd term. Let’s look at an example now.

Example 2
Find the middle term/s in the expansion of (x/2 + 3/y)9.
Solution. 
In this example, since n (= 9) is odd, we have two middle terms namely,

(9+1/2) = 5th term and (9+3/2) = 6th term.

We also have,
a = x/2, b = 3y, and n = 9.

We know that,

General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced

To find the fifth term, T5, r = 4. Therefore,

General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced


Similarly,

General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced

Therefore, the middle terms in the expansion of (x/2 + 3y)9 are 5103/16 x5y4 and 15309/8
x4y5.

More Solved Examples
Question: Find the coefficient of x6 in the expansion of (x+2)9.
Solution: 
We know that the binomial expansion of (a+b)n is,

General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced

Now, the (r + 1)th term in the expansion of (x+2)9 will be:

General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
From the above equation, we can deduce that the coefficient of the x-term is General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
Next, we need to find the coefficient of x6. Hence,
x6 = (x)9–r
Or, 6 = 9 – r, therefore r = 3.

Using this value of ‘r’,
The coefficient of x6 = General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced
= 9.8.7/1.2.3.(8) = 672.

The document General and Middle Terms in the Binomial Theorem | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on General and Middle Terms in the Binomial Theorem - Mathematics (Maths) for JEE Main & Advanced

1. What is the binomial theorem?
Ans. The binomial theorem is a formula used to expand the powers of binomials. It states that for any positive integer n, the expansion of (a + b)^n can be written as the sum of terms of the form C(n, k) * a^(n-k) * b^k, where C(n, k) represents the binomial coefficient and k ranges from 0 to n.
2. What are general terms in the binomial theorem?
Ans. The general terms in the binomial theorem refer to the individual terms obtained after expanding a binomial using the binomial theorem formula. These terms follow the pattern C(n, k) * a^(n-k) * b^k, where n represents the power of the binomial, k ranges from 0 to n, and a and b are the coefficients of the binomial.
3. What are middle terms in the binomial theorem?
Ans. The middle terms in the binomial theorem are the terms that appear in the middle of the expanded binomial when the power n is even. For example, in the expansion of (a + b)^4, the middle terms are the ones that have the power of a and b equal to 2. In general, there are (n+1)/2 middle terms in the expansion when n is even.
4. How do you find the binomial coefficient in the binomial theorem?
Ans. The binomial coefficient, denoted as C(n, k), represents the number of ways to choose k objects from a set of n objects without considering their order. It can be calculated using the formula C(n, k) = n! / (k! * (n-k)!), where "!" denotes factorial. The binomial coefficient is crucial in determining the coefficients of the terms in the binomial expansion.
5. Can the binomial theorem be applied to negative powers of binomials?
Ans. Yes, the binomial theorem can be applied to negative powers of binomials. However, to do so, the binomial coefficients must be extended to include negative values. This extension is achieved using the formula C(n, k) = (-1)^k * C(n+k-1, k), where (-1)^k accounts for the alternating signs in the expansion. Applying the binomial theorem to negative powers helps in simplifying complex algebraic expressions.
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