An infinite staircase is constructed from cubes. Find the total volume of the st
ID: 2828079 • Letter: A
Question
An infinite staircase is constructed from cubes. Find the total volume of the staircase, given that the largest cube has a side length of 2 meters and each successive cube has a side whose length is half that of the preceding cube.An infinite staircase is constructed from cubes. Find the total volume of the staircase, given that the largest cube has a side length of 2 meters and each successive cube has a side whose length is half that of the preceding cube.
An infinite staircase is constructed from cubes. Find the total volume of the staircase, given that the largest cube has a side length of 2 meters and each successive cube has a side whose length is half that of the preceding cube.
Explanation / Answer
The side lengths of the cubes are 2 , 1, 1/2, (1/2)^2, (1/2)^3, ...
So their volumes are 8 , 1, (1/2)^3, (1/2)^6, (1/2)^9.
The sum of these volumes is a geometric series with first term 8 and common ratio (1/2)^3 = 1/8. Since the sum of a geometric series is the first term divided by (1 minus common ratio), provided that the absolute value of the common ratio is less than 1, the sum of the volumes is
1/[1 - (1/8)] = 1/(7/8) = 8/7.