Question

Find the value of the following expression:
\[ 5^2 + 6^2 + 7^2 + \cdots + 20^2 = \]

• A. 2860
• B. 2840
• C. 2830
• D. 2850

Hint:

When a sum of squares does not start from \(1\), always subtract the missing initial terms after applying the standard formula.

Answer 1

The Correct Option is B

Step 1: Use the formula for sum of squares.
The formula for the sum of squares of the first \( n \) natural numbers is: \[ 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6} \]
Step 2: Find the sum of squares from \(1^2\) to \(20^2\).
\[ 1^2 + 2^2 + 3^2 + \cdots + 20^2 = \frac{20 \times 21 \times 41}{6} = 2870 \]
Step 3: Subtract the unwanted terms.
We need the sum from \(5^2\) to \(20^2\), so subtract: \[ 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30 \]
Step 4: Final calculation.
\[ 2870 - 30 = 2840 \] Final Answer: \[ \boxed{2840} \]