If $1^2 + 2^2 + 3^2 + ...... + x^2$ = $\frac{x(x+1)(2x+1)}{6}$, then the value of ($1^2 + 3^2 + 5^2 + ...... + 19^2$) is :

Q: If $1^2 + 2^2 + 3^2 + ...... + x^2$ = $\frac{x(x+1)(2x+1)}{6}$, then the value of ($1^2 + 3^2 + 5^2 + ...... + 19^2$) is :

To solve the problem of finding the value of $1^2 + 3^2 + 5^2 + \ldots + 19^2$, we need to recognize that this is a series of the squares of the first 10 odd numbers. The formula for the sum of squares of the first $n$ odd numbers is given by: \[ \text{Sum} = \frac{n(2n-1)(2n+1)}{3} \] In this problem, the sequence $1, 3, 5, \ldots, 19$ consists of the first 10 odd numbers. Therefore, $n = 10$. Substituting $n = 10$ into the formula, we get: \[ \text{Sum} = \frac{10(2 \times 10 - 1)(2 \times 10 + 1)}{3} \] Simplifying the expression: \[ = \frac{10 \times 19 \times 21}{3} \] Calculating step-by-step: 1. $10 \times 19 = 190$ 2. $190 \times 21 = 3990$ 3. $\frac{3990}{3} = 1330$ Thus, the sum of the squares of the first 10 odd numbers is 1330, which corresponds to Option 4. **Explanation of Incorrect Options:** - **Option 1 (2500):** This value is too high and does not match the calculated sum using the formula for the sum of squares of odd numbers. - **Option 2 (2485):** This value is also too high and does not correspond to the correct calculation. - **Option 3 (2100):** While closer than the previous options, this value is still incorrect as it does not match the derived sum of 1330. Therefore, based on the correct application of the formula for the sum of squares of the first $n$ odd numbers, Option 4 (1330) is indeed the correct answer.

Question

Accepted Answer

To solve the problem of finding the value of $1^2 + 3^2 + 5^2 + \ldots + 19^2$, we need to recognize that this is a series of the squares of the first 10 odd numbers. The formula for the sum of squares of the first $n$ odd numbers is given by:

$$
\text{Sum} = \frac{n(2n-1)(2n+1)}{3}
$$

In this problem, the sequence $1, 3, 5, \ldots, 19$ consists of the first 10 odd numbers. Therefore, $n = 10$.

Substituting $n = 10$ into the formula, we get:

$$
\text{Sum} = \frac{10(2 \times 10 - 1)(2 \times 10 + 1)}{3}
$$

Simplifying the expression:

$$
= \frac{10 \times 19 \times 21}{3}
$$

Calculating step-by-step:

1. $10 \times 19 = 190$
2. $190 \times 21 = 3990$
3. $\frac{3990}{3} = 1330$

Thus, the sum of the squares of the first 10 odd numbers is 1330, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1 (2500):** This value is too high and does not match the calculated sum using the formula for the sum of squares of odd numbers.
  
- **Option 2 (2485):** This value is also too high and does not correspond to the correct calculation.
  
- **Option 3 (2100):** While closer than the previous options, this value is still incorrect as it does not match the derived sum of 1330.

Therefore, based on the correct application of the formula for the sum of squares of the first $n$ odd numbers, Option 4 (1330) is indeed the correct answer.