If $ 1^2 + 2^2 + 3^2 + \dots + 10^2 = 385 $, then $ 3^2 + 6^2 + 9^2 + \dots + 30^2 $ is equal to

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Examoogle · Accepted Answer

The correct answer is Option D: 3465. Observe that the required series is $3^2(1^2 + 2^2 + ... + 10^2)$. Simply multiply the given sum ($385$) by $3^2$ ($9$).

Step-by-Step Solution & Concepts:
Shortcut Trick: Observe that the required series is $3^2(1^2 + 2^2 + ... + 10^2)$. Simply multiply the given sum ($385$) by $3^2$ ($9$).

Formula Used: $$\sum_{i=1}^{n} (k \cdot i)^2 = k^2 \cdot \sum_{i=1}^{n} i^2$$

Given: Sum of the first 10 squares is $1^{2} + 2^{2} + 3^{2} + ... + 10^{2} = 385$.

Steps: Identify the pattern: The given expression is $S_1 = 1^{2} + 2^{2} + 3^{2} + ... + 10^{2} = 385$. The target expression is $S_2 = 3^{2} + 6^{2} + 9^{2} + ... + 30^{2}$.. Factor out the common term: Factoring out $3^2$ from each term in the target expression:. Calculate the final result: Substitute the known value $385$ and calculate $9 	imes 385$.

Answer: D is correct because $3^2 + 6^2 + ... + 30^2 = 9(1^2 + 2^2 + ... + 10^2) = 9 	imes 385 = 3465$.

Exam Tip: This concept of factoring constants from squared series is frequently used in calculating variance and standard deviation in statistics.

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