Sum of the Series 1.2 2.3 3.4 ... n(n 1): A Comprehensive Guide

Understanding the sum of a series like 1.2, 2.3, 3.4, and so on, where each term is the product of two consecutive integers, is essential for various applications in mathematics and computer science. This article will guide you through the process of finding the sum of the first n terms of this series, and we'll explore different approaches to arrive at the final formula. Additionally, we'll provide a C program for this series and discuss its applicability in solving more complex problems.

Introduction to the Series

The series 1.2, 2.3, 3.4, ..., n(n 1) is a non-trivial series where each term is the product of two consecutive integers. This series is neither purely arithmetic nor purely geometric; it combines traits of both. Let's break down the problem step by step to find the sum of the first n terms.

Step-by-Step Solution

1. Identifying the General Term

The general term of this series is given by the formula:

Tn n(n 1)

2. Deriving the Sum of the Series

Now, let's find the sum of the first n terms, denoted by Sn:

Sn T1 T2 T3 ... Tn Sn 1(2) 2(3) 3(4) ... n(n 1)

This sum can be separated into two parts:

Sn (12 22 32 ... n2) (1 2 3 ... n)

We can use the known formulas for the sum of the first n natural numbers and the sum of the squares of the first n natural numbers:

Sn (12 22 32 ... n2) (1 2 3 ... n) Sn (frac{n(n 1)(2n 1)}{6}) (frac{n(n 1)}{2})

Now, let's combine these two parts into a single formula:

Sn (frac{n(n 1)(2n 1)}{6}) (frac{3n(n 1)}{6})

Sn (frac{n(n 1)(2n 1 3)}{6})

Sn (frac{n(n 1)(2n 4)}{6})

Sn (frac{n(n 1)(n 2)}{3})

Hence, the sum of the first n terms of the series is:

(boxed{frac{n(n 1)(n 2)}{3}})

C Program Implementation

Below is a C program to calculate the sum of the first n terms of the series 1.2, 2.3, 3.4, ..., n(n 1). This program allows for flexible input and displays the result in a clear manner.

#include stdio.hint main() {    int n;    float sum  0.0;    printf("Enter the number of terms (n): ");    scanf("%d", n);    for (int i  1; i 

Related Keywords and Applications

Sum of series: This keyword is crucial for understanding the various methods to calculate the sum of different types of series, including our current series. It encompasses problems involving arithmetico-geometric series and harmonic series, among others. Arithmetic and geometric progression: While our series is neither purely arithmetic nor purely geometric, the understanding of these two progressions forms the foundation for solving many types of series problems, including the one discussed here. Sum of n terms: This is a broader keyword that covers the general concept of finding the sum of a sequence of numbers, whether they follow a pattern or not. It is frequently used in mathematical and computer science problems involving sequence and series.

Mastering the sum of such series can help in solving more complex problems in mathematics and computer science. Understanding and implementing the solution in a programming language such as C can further enhance your knowledge and practical skills.