General term in the expansion of $(x+a)^n$ is :

Q: General term in the expansion of $(x+a)^n$ is :

The general term in the expansion of $(x+a)^n$ is given by the binomial theorem, which states that: \[ (x+a)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} a^r \] The general term, often denoted as $T_{r+1}$, is: \[ T_{r+1} = \binom{n}{r} x^{n-r} a^r \] This matches exactly with Option 1: $^nC_r x^{n-r} a^r$. ### Explanation for Option 1: - **$^nC_r$**: This represents the binomial coefficient, which is the number of ways to choose $r$ elements from a set of $n$ elements. It is a crucial part of the binomial expansion formula. - **$x^{n-r}$**: This term indicates that the power of $x$ decreases as $r$ increases, starting from $n$ and going down to 0. - **$a^r$**: This term shows that the power of $a$ increases as $r$ increases, starting from 0 and going up to $n$. ### Explanation for why other options are incorrect: - **Option 2: $^nC_r x^r a^r$** - This option incorrectly assigns the same power $r$ to both $x$ and $a$. In the binomial expansion, the powers of $x$ and $a$ should add up to $n$, which is not the case here. - **Option 3: $^nC_{n-r} x^{n-r} a^r$** - While the powers of $x$ and $a$ are correctly assigned, the binomial coefficient $^nC_{n-r}$ is incorrect. The correct coefficient should be $^nC_r$, as the binomial coefficient is based on the number of terms chosen, not the remaining terms. - **Option 4: $^nC_{n-r} x^r a^{n-r}$** - This option incorrectly swaps the powers of $x$ and $a$. The power of $x$ should decrease from $n$ to 0, and the power of $a$ should increase from 0 to $n$. Additionally, the binomial coefficient $^nC_{n-r}$ is incorrect for the same reason as in Option 3. In conclusion, Option 1 is the correct representation of the general term in the expansion of $(x+a)^n$ according to the binomial theorem.

Question

Accepted Answer

The general term in the expansion of $(x+a)^n$ is given by the binomial theorem, which states that:

$$
(x+a)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} a^r
$$

The general term, often denoted as $T_{r+1}$, is:

$$
T_{r+1} = \binom{n}{r} x^{n-r} a^r
$$

This matches exactly with Option 1: $^nC_r x^{n-r} a^r$.

### Explanation for Option 1:

- **$^nC_r$**: This represents the binomial coefficient, which is the number of ways to choose $r$ elements from a set of $n$ elements. It is a crucial part of the binomial expansion formula.
- **$x^{n-r}$**: This term indicates that the power of $x$ decreases as $r$ increases, starting from $n$ and going down to 0.
- **$a^r$**: This term shows that the power of $a$ increases as $r$ increases, starting from 0 and going up to $n$.

### Explanation for why other options are incorrect:

- **Option 2: $^nC_r x^r a^r$**
  - This option incorrectly assigns the same power $r$ to both $x$ and $a$. In the binomial expansion, the powers of $x$ and $a$ should add up to $n$, which is not the case here.

- **Option 3: $^nC_{n-r} x^{n-r} a^r$**
  - While the powers of $x$ and $a$ are correctly assigned, the binomial coefficient $^nC_{n-r}$ is incorrect. The correct coefficient should be $^nC_r$, as the binomial coefficient is based on the number of terms chosen, not the remaining terms.

- **Option 4: $^nC_{n-r} x^r a^{n-r}$**
  - This option incorrectly swaps the powers of $x$ and $a$. The power of $x$ should decrease from $n$ to 0, and the power of $a$ should increase from 0 to $n$. Additionally, the binomial coefficient $^nC_{n-r}$ is incorrect for the same reason as in Option 3.

In conclusion, Option 1 is the correct representation of the general term in the expansion of $(x+a)^n$ according to the binomial theorem.