A sequence where each bottle holds half the previous, starting with 729, and we want the number of terms until the volume filled (sum of capacities) exceeds or equals the total solution 729. But sum $S_n = 729(1 - 2^{-n}) < 729$, never reaches.

February 22, 2026

$A sequence where each bottle holds half the previous, starting with 729, and we want the number of terms until the volume filled (sum of capacities) exceeds or equals the total solution 729. But sum $S_n = 729(1 - 2^{-n}) < 729$, never reaches.$

Not helpful.
Instead, question is: How many bottles are needed until the *sum of capacities* reaches 729? But total capacity is infinite.
Given context, likely intended:
This is not working.
Correct and intended interpretation from similar problems:
The total volume 729 liters is poured into bottles, each with capacity half the previous, starting with capacity C. We want how many bottles to hold exactly 729.

A sequence where each bottle holds half the previous, starting with 729, and we want the number of terms until the volume filled (sum of capacities) exceeds or equals the total solution 729. But sum \(S_n = 729(1 - 2^{-n}) < 729\), never reaches.

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