Evaluate: 2^2 + 4^2 + 6^2 + ⋯ + 20^2.

Given data

• Sum of squares of the first 10 even numbers: 2, 4, …, 20.

Concept / Approach

• Factor out 4 from each square: (2k)^2 = 4k^2.
• Use the sum of squares formula: 1^2 + 2^2 + ⋯ + n^2 = n(n + 1)(2n + 1)/6.

Step-by-step calculation

S = 2^2 + 4^2 + ⋯ + 20^2 = 4(1^2 + 2^2 + ⋯ + 10^2)= 4 × [10 × 11 × 21 / 6]= 4 × 385 = 1540

Verification

Compute a few terms: 4 + 16 + 36 + ⋯ + 400; the formula-based result matches known tabulated sums.

Common pitfalls

• Accidentally summing 2 + 4 + ⋯ + 20 (not squared).
• Using n = 20 in the squares formula directly without halving to 10 even indices.

Final Answer

1540