A Billionth Portion of Infinity?

📌 CONCEPT: The concept of an infinite series and sequence is related to the idea of an infinite number of terms or elements, where each term is divided into two equal parts, and the amount of water left in the original cup is a result of this infinite process.

📐 RULE / FORMULA: The rule or formula that applies here is the sum of an infinite geometric series, which is given by S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

💡 WORKED EXAMPLE: If you start with a cup that's half full (a = 1) and each portion is poured out completely, and you divide each portion into two equal parts (r = 1/2), then the amount of water left in the original cup is S = 1 / (1 - 1/2) = 1 / (1/2) = 2.

⚠️ COMMON MISTAKE: Students often get wrong the idea of an infinite series and sequence, thinking that it means an infinite number of terms, but in reality, it means the sum of an infinite number of terms, which can be calculated using the formula for the sum of an infinite geometric series.