Given that $1^2 + 2^2 + 3^2 + ..... + 10^2 = 385$, then find the value of $2^2 + 4^2 + 6^2 + ...... + 20^2$.

### Concept & Formula The problem requires finding the sum of a modified series based on a given standard series of squares. The key insight is factoring out a common multiplier from the target series. $$ (ab)^2 = a^2 \times b^2 $$ ### Step-by-Step Solution **Given:** $1^2 + 2^2 + 3^2 + ... + 10^2 = 385$ **Calculation:** We need to find the value of: $2^2 + 4^2 + 6^2 + ... + 20^2$ Rewrite each term by factoring out $2$ from the base: $= (2 \times 1)^2 + (2 \times 2)^2 + (2 \times 3)^2 + ... + (2 \times 10)^2$ Apply the exponent rule $(ab)^2 = a^2b^2$: $= (2^2 \times 1^2) + (2^2 \times 2^2) + (2^2 \times 3^2) + ... + (2^2 \times 10^2)$ Factor out the common term $2^2$ (which is $4$): $= 4 \times (1^2 + 2^2 + 3^2 + ... + 10^2)$ Substitute the given value of the original series ($385$): $= 4 \times 385 = 1540$ ### Exam Strategy & Shortcut When you see a series where each base is a direct multiple of the base in a known series of squares, simply square that multiple and multiply it by the given sum. Here, the bases are doubled ($2x$), so the total sum increases by a factor of $2^2 = 4$. Mentally calculate $4 \times 385 = 1540$. ### Common Pitfall A frequent mistake is factoring out $2$ instead of $2^2$, leading students to calculate $2 \times 385 = 770$. Remember that when extracting a factor from a squared term, the extracted factor must also be squared. ### Final Answer Therefore, the correct answer is **1540**.