Question

Martin made a stick pattern such that each successive stick is twice as long as the one before it. If the shortest stick in the pattern is long, and there are sticks in total, which is closest to the combined length of the sticks in Martin's pattern?

Answer 1

The combined length of the sticks in Martin's pattern is equal to the sum of a geometric sequence with first term equals to the length of the shortest stick (let's call this length "a") and common ratio equal to 2. The formula for the sum of the first n terms of a geometric sequence with first term a and common ratio r is:

S_n = a(1 - r^n) / (1 - r)

In this case, we have n sticks in total, so we need to calculate S_n with n:

S_n = a(1 - 2^n) / (1 - 2)

S_n = a(2^n - 1)

Therefore, the combined length of the sticks in Martin's pattern is closest to (2^n - 1) multiplied by the length of the shortest stick.