Geometric Progression

Let the digits of the number be , , and , where , , and are in geometric progression. This gives: Now, according to the given conditions: 1. The sum of the right hand and left hand digits exceeds twice the middle digit by 1: 2. The sum of the left hand and middle digits is two third of the sum of the middle and right hand digits: Now we can solve these equations
step by step.
From the second equation: Multiplying both sides by 3: From the first equation: Substitute Equation 2 into Equation 1: Now use Equation 2 and substitute values to find the sum of the digits. Solving the system of equations will give: Thus, the sum of the digits .

The sum of an infinite geometric series with first term and common ratio (where ) is given by the formula: We are given that the sum of the infinite series is 4, so: (Equation 1) We are also given that the second term of the series is . The second term is given by: (Equation 2) From Equation 2, we can express as: Now, substitute this value of into Equation 1: Multiply both sides by : Expanding the right-hand side: Rearranging the terms: This is a quadratic equation in . Using the quadratic formula: So, we have two possible values for : or Now, substituting into Equation 2: Thus, the correct answer is and .