Sum of a Special Arithmetic-Geometric Sequence: A Comprehensive Guide

Welcome to this comprehensive guide on finding the sum of a special sequence that combines arithmetic and geometric progression principles. The sequence in question is defined as: n 1/10^1, (n-1) 1/10^2, ..., 2 1/10^(n-1), 1 1/10^n.

Understanding the Summation

The given sum can be mathematically represented as:

S n 1/10^1 (n-1) 1/10^2 ... 2 1/10^(n-1) 1 1/10^n

Multiplying with the Common Ratio

To solve this, we multiply the series by its common ratio, which is 1/10:

S  n 1/10^1  (n-1) 1/10^2  ...  2 1/10^(n-1)  1 1/10^nS(1/10)  n 1/10^2  (n-1) 1/10^3  ...  2 1/10^n  1 1/10^n (1)

Next, we subtract equation (1) from equation (2):

1 - S(1/10)  n 1/10^1 - (1/10^2   1/10^3   ...   1/10^n) - 1/10^n

Summation of Geometric Series

The series inside the parentheses is a geometric series with the common ratio of 1/10 and the first term is 1/10^2. Using the formula for the sum of a geometric series:

Sum a(1 - r^n) / (1 - r)

We can find the sum as:

Sum  (1/10^2) (1 - (1/10)^n) / (1 - 1/10)Sum  (1/100) (1 - (1/10)^n) / (9/10)Sum  (1/100) (1 - (1/10)^n) / (9/10)  (1/90) (1 - (1/10)^n)

Substituting the sum back into the original equation:

S - (1/90) (1 - (1/10)^n)  -n (1/10) (1/90) (1 - (1/10)^n)S  -n / 9 (1 - (1/10)^n)

A Different Approach: Splitting the Sum

We can split the sum into two parts with x 1/10:

x * (1   2x   3x^2   ...   nx^(n-1)) - (x^2   2x^3   3x^4   ...   (n-1)x^n)

The first part is just a geometric series multiplied by n, and the second part can be evaluated using the GP summation formula.

Rational Polynomial Function

Let's define a rational polynomial function:

fn(x)  sum_{i1}^n i (1/x)^(n-1-i)

This is the same as the given sequence:

fn(10)

Now, we manipulate the sum:

fn(x)  sum_{i1}^n i (1/x)^(n-1-i)  x^(-n) sum_{i1}^n i (x^(i-1))  x^(-n) sum_{i1}^n (d/dx x^i)

We can write it as:

fn(x)  x^(-n) sum_{i1}^n (d/dx x^i)  x^(-n) (d/dx sum_{i1}^n x^i)  x^(-n) (d/dx (1 - x^n / 1 - x))  (nx^(n-1) - nx^n - 1) / (x^n - 1 - x^2)  (nx - n - 1) / (1 - x^2) * (1 / (x^n - 1 - x^2))

Substitute x 1/10 to get the final answer:

S  (n/10 - n - 1) / (1 - 1/100) * (1 / (1/10^n - 1 - 1/100))