Question

Given that, $1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 165$, then what is the value of $3^2 + 9^2 + 15^2 + 21^2 + 27^2$?

• A. \(1485 \)
• B. \(1385 \)
• C. \(990 \)
• D. \(495 \)
• E. \(995 \)

Hint:

If each term is multiplied by a constant $k$, then: \[ (kx)^2 = k^2 x^2 \] So factor out $k^2$ from the entire sum.

Answer 1

The Correct Option is A

Concept: Observe that: \[ 3, 9, 15, 21, 27 = 3 \times (1, 3, 5, 7, 9) \] So each term is 3 times the corresponding term in the given expression.
Step 1: Express in terms of given sequence.
\[ 3^2 + 9^2 + 15^2 + 21^2 + 27^2 = (3 \cdot 1)^2 + (3 \cdot 3)^2 + (3 \cdot 5)^2 + (3 \cdot 7)^2 + (3 \cdot 9)^2 \] \[ = 3^2(1^2 + 3^2 + 5^2 + 7^2 + 9^2) \]

Step 2: Substitute given value.
\[ = 9 \times 165 \]

Step 3: Compute the result.
\[ = 1485 \]