The coefficient of $x^{2}$ in the expansion of $(20x^{2}-9x+1)^{-1}is:$ (Here $|x|

Question

The coefficient of $x^{2}$ in the expansion of $(20x^{2}-9x+1)^{-1}is:$ (Here $|x|<\frac{1}{5})$

Accepted Answer

To find the coefficient of $x^2$ in the expansion of $(20x^2 - 9x + 1)^{-1}$, we can use the binomial series expansion for a function of the form $(1 + u)^{-1}$, which is given by:

$$
(1 + u)^{-1} = 1 - u + u^2 - u^3 + \cdots
$$

In this case, we need to express $(20x^2 - 9x + 1)^{-1}$ in a form suitable for applying the binomial series. We rewrite the expression as:

$$
(1 - (9x - 20x^2))^{-1}
$$

Here, $u = 9x - 20x^2$. According to the binomial series expansion, we have:

$$
(1 + u)^{-1} = 1 - (9x - 20x^2) + (9x - 20x^2)^2 - \cdots
$$

We are interested in the coefficient of $x^2$. Let's expand the terms:

1. The first term is $1$, which does not contribute to the $x^2$ term.

2. The second term is $-(9x - 20x^2)$, which contributes $-20x^2$.

3. The third term is $(9x - 20x^2)^2$. Expanding this, we get:

$$
(9x - 20x^2)^2 = (9x)^2 - 2(9x)(20x^2) + (20x^2)^2
$$

Calculating each part:

- $(9x)^2 = 81x^2$
- $-2(9x)(20x^2) = -360x^3$ (does not contribute to $x^2$)
- $(20x^2)^2 = 400x^4$ (does not contribute to $x^2$)

Thus, the contribution to the $x^2$ term from $(9x - 20x^2)^2$ is $81x^2$.

Adding the contributions to the $x^2$ term from the expansion:

- From $-(9x - 20x^2)$, we have $-20x^2$.
- From $(9x - 20x^2)^2$, we have $81x^2$.

The total coefficient of $x^2$ is:

$$
-20 + 81 = 61
$$

Therefore, the coefficient of $x^2$ in the expansion is $61$, which corresponds to Option 3.

**Explanation of Incorrect Options:**

- **Option 1 (-20):** This is only the contribution from the linear term $-(9x - 20x^2)$ and does not account for the quadratic term $(9x - 20x^2)^2$.

- **Option 2 (20):** This does not match any logical calculation from the expansion process.

- **Option 4 (60):** This is close but incorrect, as it does not accurately sum the contributions from both the linear and quadratic terms.

Thus, Option 3 is the correct answer, as it accurately reflects the total contribution to the $x^2$ term in the expansion.