How many men paid for a total amount of $262,143.75 if the first person to arrive only paid 25 cents?

Problem
In a fund raising show, a group of philanthropists agreed that the first one to arrive would pay 25¢ to enter, and each later would pay twice as much as the preceding person. The total amount collected from all of them was \$262,143.75. How many of them paid?

Derivation of Sum of Finite and Infinite Geometric Progression

Geometric Progression, GP
Geometric progression (also known as geometric sequence) is a sequence of numbers where the ratio of any two adjacent terms is constant. The constant ratio is called the common ratio, r of geometric progression. Each term therefore in geometric progression is found by multiplying the previous one by r.
 

Eaxamples of GP:

  • 3, 6, 12, 24, … is a geometric progression with r = 2
  • 10, -5, 2.5, -1.25, … is a geometric progression with r = -1/2

 

Derivation of Sum of Arithmetic Progression

Arithmetic Progression, AP
Definition

Arithmetic Progression (also called arithmetic sequence), is a sequence of numbers such that the difference between any two consecutive terms is constant. Each term therefore in an arithmetic progression will increase or decrease at a constant value called the common difference, d.
 

Examples of arithmetic progression are:

  • 2, 5, 8, 11,... common difference = 3
  • 23, 19, 15, 11,... common difference = -4