What is the value of `16^(2) +17^(2) +18^(2) + …….25^(2)`?

Options

A. 4325
B. 4465
C. 4105
D. 4285

Answer

To find the value of \( 16^2 + 17^2 + 18^2 + \ldots + 25^2 \), we can use the formula for the sum of squares of the first \( n \) natural numbers:

\[ \text{Sum of squares} = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 1: Calculate the sum of squares from 1 to 25
We first calculate the sum of squares from \( 1^2 \) to \( 25^2 \).

Using the formula with \( n = 25 \):

\[ \text{Sum}_{1 \text{ to } 25} = \frac{25(25 + 1)(2 \cdot 25 + 1)}{6} \]

Calculating this step by step:

- \( 25 + 1 = 26 \)
- \( 2 \cdot 25 + 1 = 51 \)

Now substituting these values into the formula:

\[ \text{Sum}_{1 \text{ to } 25} = \frac{25 \cdot 26 \cdot 51}{6} \]

Calculating \( 25 \cdot 26 = 650 \):

\[ \text{Sum}_{1 \text{ to } 25} = \frac{650 \cdot 51}{6} \]

Now calculating \( 650 \cdot 51 = 33150 \):

\[ \text{Sum}_{1 \text{ to } 25} = \frac{33150}{6} = 5525 \]

### Step 2: Calculate the sum of squares from 1 to 15
Next, we calculate the sum of squares from \( 1^2 \) to \( 15^2 \).

Using the formula with \( n = 15 \):

\[ \text{Sum}_{1 \text{ to } 15} = \frac{15(15 + 1)(2 \cdot 15 + 1)}{6} \]

Calculating this step by step:

- \( 15 + 1 = 16 \)
- \( 2 \cdot 15 + 1 = 31 \)

Now substituting these values into the formula:

\[ \text{Sum}_{1 \text{ to } 15} = \frac{15 \cdot 16 \cdot 31}{6} \]

Calculating \( 15 \cdot 16 = 240 \):

\[ \text{Sum}_{1 \text{ to } 15} = \frac{240 \cdot 31}{6} \]

Now calculating \( 240 \cdot 31 = 7440 \):

\[ \text{Sum}_{1 \text{ to } 15} = \frac{7440}{6} = 1240 \]

### Step 3: Calculate the sum from 16 to 25
Now we can find the sum from \( 16^2 \) to \( 25^2 \) by subtracting the two sums:

\[ \text{Sum}_{16 \text{ to } 25} = \text{Sum}_{1 \text{ to } 25} - \text{Sum}_{1 \text{ to } 15} \]

Substituting the values we calculated:

\[ \text{Sum}_{16 \text{ to } 25} = 5525 - 1240 = 4285 \]

### Final Answer
Thus, the value of \( 16^2 + 17^2 + 18^2 + \ldots + 25^2 \) is \( \boxed{4285} \).