Binomial Theorem Formula – General Term
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 T_{1}, T_{2}, T_{3}, T_{4}, … T_{n+1} are the first, second, third, fourth, … (n + 1)th terms, respectively in the expansion of (a+b)^{n}. Therefore,
Generalizing it, we have the formula for the general term:
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
To find the fourth term, T_{4}, r = 3. Therefore,
Hence, the fourth term in the expansion of (3x–y)^{7} = – 2835x^{4}y^{3}
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,
To find the fifth term, T_{5}, r = 4. Therefore,
Similarly,
Therefore, the middle terms in the expansion of (x/2 + 3y)^{9} are 5103/16 x^{5}y^{4} and 15309/8
x^{4}y^{5}.
More Solved Examples
Question: Find the coefficient of x^{6} in the expansion of (x+2)^{9}.
Solution: We know that the binomial expansion of (a+b)^{n} is,
Now, the (r + 1)th term in the expansion of (x+2)^{9} will be:
From the above equation, we can deduce that the coefficient of the xterm is
Next, we need to find the coefficient of x6. Hence,
x^{6} = (x)^{9–r}
Or, 6 = 9 – r, therefore r = 3.
Using this value of ‘r’,
The coefficient of x^{6} =
= 9.8.7/1.2.3.(8) = 672.
209 videos443 docs143 tests

1. What is the binomial theorem? 
2. What are general terms in the binomial theorem? 
3. What are middle terms in the binomial theorem? 
4. How do you find the binomial coefficient in the binomial theorem? 
5. Can the binomial theorem be applied to negative powers of binomials? 
209 videos443 docs143 tests


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