The requested paper strip division into 500 parts is not possible using the method of successively dividing the largest part into 7 parts because it would involve fractional cuts, which cannot happen in a physical scenario.
The student has asked if it is possible to end up with 500 parts after successively dividing the largest part of a paper strip into 7 parts on each step. To figure out if it is possible to reach exactly 500 parts by following this procedure, let's analyze the problem using the concept of geometric progressions and the properties of numbers.
When we divide the strip for the first time, we get 7 parts. If we denote the initial piece as one part, then after the first cut we have 7 pieces. After the second cut, we also increase the total number of parts by 6 additional parts (one part is divided into 7, so it's 7 new minus the 1 that was divided). So with each cut, we are adding 6 additional parts to the total count.
If we start with 1 part, then after the first cut, we will have:
- Initial part (1)
- + 6 (because we cut the largest part into 7 pieces)
This can be represented by a series where each term is 6 more than the previous term, starting from 1 (the initial part):
1, 7, 13, 19, 25, ...
This series can be represented by the formula: 1 + 6n, where n is the number of cuts made.
To find out if we can get to 500 pieces, we need to solve for n in the equation:
1 + 6n = 500
Subtracting 1 from both sides gives us:
6n = 499
If we divide both sides by 6:
n = 499 / 6
n = 83.166...
Because n must be a whole number (you can't have a fraction of a cut), and since 499 is not divisible by 6, it is impossible to end up with exactly 500 parts by repeatedly cutting the largest part into 7 pieces.