Problem Description

Given is the arithmetic-geometric sequence u(k) = (900-3k)r^k-1.
Let s(n) = Σ_k=1…nu(k).

Find the value of r for which s(5000) = -600,000,000,000.

Give your answer rounded to 12 places behind the decimal point.

Analysis

Simple problem using a binary search and keep reducing the range until we reach the target. The first thing we did was divide the equation by 3 as that seemed obvious and thought it might help speed things up. So the target became -200,000,000,000 and the equation (300-k) * r**(k-1). Did save a multiplication, but wasn’t really necessary.

Solution

Runs < 1 second in Python.

sn = 5000
sm = -2 * 10**11
s, r, d = 0, 1, 0.1
while abs(s – sm) > 1:
  s = sum((300 – k) * r**(k-1) for k in range(1, sn+1))
  r += d if s>sm else -d
  d /= 2

print ‘Answer to PE 235 = %.12f’ % r

Comments

Worked first time!