Question

The sum of the consecutive integers \(1,2,3, \ldots, n\) is given by the formula \(\frac{1}{2} n(n+1) .\) How many consecutive integers, starting with \(1,\) must be added to get a sum of \(703 ?\)

Short answer

37

Step-by-step solution

- Understanding the Formula

The sum of the first n consecutive integers starting from 1 is given by the formula \ \( S = \frac{1}{2} n(n+1) \). This means if you want to find out how many integers starting from 1 add up to a sum S, you need to solve for n in the equation \ \( \frac{1}{2} n(n+1) = S \).

- Setting up the Equation

In this problem, the sum of the integers is given as 703. So, we set up the equation: \ \( \frac{1}{2} n(n+1) = 703 \).

- Simplifying the Equation

Multiply both sides of the equation by 2 to get rid of the fraction: \ \( n(n+1) = 1406 \).

- Solving the Quadratic Equation

You now have a quadratic equation: \ \( n^2 + n - 1406 = 0 \). Solve for n using the quadratic formula: \ \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \ \( a=1, b=1, \) and \ \( c=-1406 \).

- Calculating the Values

Substitute the values of a, b, and c into the quadratic formula: \ \( n = \frac{-1 \pm \sqrt{1 + 4 \cdot 1406}}{2} = \frac{-1 \pm \sqrt{1 + 5624}}{2} = \frac{-1 \pm \sqrt{5625}}{2} \).

- Determining the Positive Solution

Calculate \ \( \sqrt{5625} = 75 \), hence we have: \ \( n = \frac{-1 + 75}{2} = \frac{74}{2} = 37 \). The other solution is negative, so we discard it.

Key concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in mathematics. It has the form \( ax^2 + bx + c = 0 \, \text{where} \, a eq 0 \). In the example given, the equation emerges when solving for the number of consecutive integers that sum to a specific value. Here is how it looks in the problem context: \( n^2 + n - 1406 = 0 \). This type of equation is called 'quadratic' because it contains the variable raised to the power of two.

Algebraic Formula

An algebraic formula is a mathematical statement that uses algebraic expressions to represent a relationship. In this exercise, the relevant algebraic formula is: \( \frac{1}{2} n(n+1) \). This formula calculates the sum of the first n consecutive integers starting from 1. It’s an efficient tool to find sums quickly without needing to add individual numbers.

Consecutive Numbers

Consecutive numbers are integers that follow each other in order. Each number is one more than the previous number. For example, \( 1, 2, 3, 4, \ldots \) are consecutive numbers. In this exercise, the focus was on finding how many consecutive integers starting from 1 add up to 703 using the given formula. Understanding consecutive numbers helps in identifying patterns and understanding the concept of sequence.