Question

$1^3 - 2^3 + 3^3 - 4^3 + \cdots + 9^3 =$

• A. $425$
• B. $-425$
• C. $475$
• D. $-475$

Hint:

In alternating series, pairing consecutive terms often simplifies calculations.

Answer 1

The Correct Option is A

Step 1: Group the terms in pairs: \[ (1^3-2^3) + (3^3-4^3) + (5^3-6^3) + (7^3-8^3) + 9^3 \]
Step 2: Evaluate each pair: \[ 1^3-2^3 = 1-8 = -7 \] \[ 3^3-4^3 = 27-64 = -37 \] \[ 5^3-6^3 = 125-216 = -91 \] \[ 7^3-8^3 = 343-512 = -169 \]
Step 3: Add all negative terms: \[ -7-37-91-169 = -304 \]
Step 4: Add the remaining term: \[ 9^3 = 729 \]
Step 5: Find the total sum: \[ 729 - 304 = 425 \]