This article explores the pattern of 1*2+2*3+3*4+…+n(n+1). It will explain how to understand the pattern and how to calculate the result.

## Understanding the Pattern

The pattern of 1*2+2*3+3*4+…+n(n+1) is a sequence of numbers that can be expressed as a mathematical equation. Each number in the sequence is the product of the two numbers that precede it. The first number in the sequence is 1 and the last number is n multiplied by n plus 1.

For example, if the sequence is 1*2+2*3+3*4+4*5+5*6, then the first number is 1, the second number is 2, the third number is 3, the fourth number is 4, and the fifth number is 5 multiplied by 6. The pattern can be repeated for any number of terms.

## Calculating the Result

To calculate the result of 1*2+2*3+3*4+…+n(n+1), the first step is to determine the number of terms in the sequence. This can be done by counting the number of terms in the equation.

Once the number of terms is known, the result can be calculated by multiplying each number in the sequence by the one that follows it and then adding all the products together. For example, if the sequence is 1*2+2*3+3*4+4*5+5*6, then the result would be 1*2+2*3+3*4+4*5+5*6 = 1+6+12+20+30 = 69.

In conclusion, the pattern of 1*2+2*3+3*4+…+n(n+1) is a sequence of numbers that can be expressed as a mathematical equation. To calculate the result, the number of terms must be determined and then each number in the sequence must be multiplied by the one that follows it and the products added together.

This mathematical equation, 1 x 2 + 2 x 3 + 3 x 4 +…+ n (n + 1), is a particular way of expressing mathematical concepts and summations. It is important to understand this equation in order to solve many mathematical problems.

This equation can be identified as the general term of a geometric series. It is used to find the sum of a finite number of terms. This is done by multiplying the first term (1) with the common difference (2) and summing the terms up to the final one. The resulting sum is the sum of the series.

For example, if n is 6, the equation looks like this: 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + 5 x 6 + 6 x 7 = _. To solve it, calculate the left hand side of the equation (84) and add the last product of n and n + 1 (42). As a result, the answer is 126.

This equation can also be calculated algebraically. To do so, find the sum of the first term and the last term, multiply the sum by the total number of terms, and divide the product by two. In the example above, (1 + 7) x 6 would be 48, and 48 divided by two is 24. This approach yields the same answer as the first method.

In conclusion, understanding this equation is essential to applying numerous mathematical concepts. While it can be solved in multiple ways, the equation presented above provides skilled mathematicians with a way of quickly evaluating the sum of a series.