Question

Find the sum $$1 \cdot 2+2 \cdot 3+\cdots+(n-1) n$$.

   Find the sum
$$1 \cdot 2+2 \cdot 3+\cdots+(n-1) n$$.

Discrete Mathematics

Discrete Mathematics

Richard Johnsonbaugh 8th Edition

Chapter 6, Problem 21

Instant Answer

Solved by Expert Prabhat Tyagi

12/11/2022

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Step 1: We are given the sum of the series as: $$1 \cdot 2+2 \cdot 3+\cdots+(n-1) n$$ Show more…

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Find the sum $$1 \cdot 2+2 \cdot 3+\cdots+(n-1) n$$.

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Key Concepts

Summation Notation

This concept involves using sigma notation to concisely represent a series of terms indexed by an integer. It allows for compact representation and manipulation of sums, transforming a long sequence of additions into a single, general expression that can be analyzed and simplified.

Summation Formulas

Familiarity with standard summation formulas, such as the formula for the sum of the first n natural numbers and the sum of the first n squares, is essential for converting a complex series into a closed-form expression. These formulas enable one to break down and compute sums efficiently without having to add each term individually.

Algebraic Manipulation

This refers to the process of rewriting and decomposing expressions to simplify the evaluation of a series. By recognizing that a product, like k(k+1), can be split into separate sums (for instance, k^2 and k), one can apply known summation formulas and combine the results to derive a compact algebraic expression for the series.

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