Given that 1^2 + 2^2 + 3^2 + ldots + 10^2 = 385, what is the value of 3^2 +...

1. Translate the problem requirements: We need to find the sum of squares of multiples of 3 from \(3^2\) to \(30^2\). We're given the sum of squares from \(1^2\) to \(10^2\) equals 385.
2. Identify the pattern relationship: Recognize that \(3^2, 6^2, 9^2, \ldots, 30^2\) can be written as \((3\times1)^2, (3\times2)^2, (3\times3)^2, \ldots, (3\times10)^2\).
3. Apply the multiplication property of squares: Use the fact that \((3n)^2 = 9n^2\) to factor out the common multiplier from each term.
4. Calculate using the given information: Substitute the known sum to find our answer directly.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're looking for. We want to find the value of \(3^2 + 6^2 + 9^2 + \ldots + 30^2\). Notice that these are all multiples of 3: we're looking at 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30. That's 10 numbers total, and they're all multiples of 3.

We're given that \(1^2 + 2^2 + 3^2 + \ldots + 10^2 = 385\). This is the sum of squares of the first 10 positive integers.

Process Skill: TRANSLATE - Converting the problem statement into clear mathematical understanding

2. Identify the pattern relationship

Now let's look at the connection between what we know and what we want to find. The numbers we want to square are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30.

If we factor out 3 from each number, we get:

• \(3 = 3 \times 1\)
• \(6 = 3 \times 2\)
• \(9 = 3 \times 3\)
• \(12 = 3 \times 4\)
• \(15 = 3 \times 5\)
• \(18 = 3 \times 6\)
• \(21 = 3 \times 7\)
• \(24 = 3 \times 8\)
• \(27 = 3 \times 9\)
• \(30 = 3 \times 10\)

So our sum becomes: \((3\times1)^2 + (3\times2)^2 + (3\times3)^2 + \ldots + (3\times10)^2\)

Process Skill: INFER - Recognizing the non-obvious pattern that connects the two sums

3. Apply the multiplication property of squares

Here's the key insight: when we square a number that's been multiplied by 3, we can factor out that multiplication.

For any number like \(3\times1\), when we square it: \((3\times1)^2 = 3^2 \times 1^2 = 9 \times 1^2\)

This works for all our terms:

• \((3\times1)^2 = 9 \times 1^2\)
• \((3\times2)^2 = 9 \times 2^2\)
• \((3\times3)^2 = 9 \times 3^2\)
• and so on...

So our entire sum becomes:
\(3^2 + 6^2 + 9^2 + \ldots + 30^2 = 9\times1^2 + 9\times2^2 + 9\times3^2 + \ldots + 9\times10^2\)

We can factor out the 9:
\(= 9 \times (1^2 + 2^2 + 3^2 + \ldots + 10^2)\)

4. Calculate using the given information

Now we can substitute what we know! We're told that \(1^2 + 2^2 + 3^2 + \ldots + 10^2 = 385\).

Therefore:
\(3^2 + 6^2 + 9^2 + \ldots + 30^2 = 9 \times (1^2 + 2^2 + 3^2 + \ldots + 10^2)\)
\(= 9 \times 385\)
\(= 3,465\)

Final Answer

The value of \(3^2 + 6^2 + 9^2 + \ldots + 30^2\) is 3,465.

Looking at our answer choices, this matches choice E: 3,465.

Common Faltering Points

Errors while devising the approach

1. Missing the pattern connection between the two sequences

Students often fail to recognize that 3, 6, 9, ..., 30 are all multiples of 3 times the sequence 1, 2, 3, ..., 10. They might try to calculate each square individually or look for a completely different approach, missing the elegant relationship that allows them to use the given information.

2. Misunderstanding what needs to be calculated

Students may miscount the terms in the sequence \(3^2 + 6^2 + 9^2 + \ldots + 30^2\). They might think there are fewer or more terms than the actual 10 terms, leading them to set up an incorrect relationship with the given sum of \(1^2 + 2^2 + \ldots + 10^2\).

3. Attempting to use sum of squares formula incorrectly

Some students might try to apply the formula for sum of squares of first n natural numbers directly to the sequence 3, 6, 9, ..., 30 without recognizing that this sequence is not consecutive natural numbers, leading to an incorrect setup.

Errors while executing the approach

1. Incorrectly applying the square of a product rule

When squaring terms like \((3\times1)\), students might write \((3\times1)^2 = 3 \times 1^2\) instead of the correct \(3^2 \times 1^2 = 9 \times 1^2\). This leads to factoring out 3 instead of 9, resulting in \(3 \times 385 = 1,155\) instead of the correct answer.

2. Arithmetic errors in final multiplication

Students may correctly identify that the answer is \(9 \times 385\) but make computational errors when multiplying, such as calculating \(9 \times 385 = 3,365\) or 3,485 instead of 3,465.

Errors while selecting the answer

1. Choosing the wrong multiple of 385

Students who incorrectly factor out 3 instead of 9 will get 1,155 (which is \(3 \times 385\)) and select answer choice A. This is a common error because 1,155 appears as a tempting answer choice that results from the most common computational mistake.