Finding the Sum of the Arithmetic Sequence

An arithmetic sequence is a sequence of numbers such that the difference between two consecutive terms is constant.  The sequence

7, 13, 19, 25, 31, 37, 43, 59

is an example of an arithmetic sequence with first term 7, constant difference 6, and last term 49.

You have learned in that the formula for finding the nth term of the arithmetic sequence a_n with first term a_1, and constant difference d is given by

a_n = a_1 + (n-1)d .

In this post, we derive the formula for finding the sum of all the numbers in an arithmetic sequence. We take the specific example above and use Gauss’ method in finding the sum of the first 100 positive integers. Recall that in adding the first 100 integers, Gauss added the first integer to the last, the second integer to  the second to the last, the third integer and the third to the last and so on. » Read more

Equation of a line: The derivation of y = mx + b

We have discussed in context the origin (click here and here) of the linear equation y = ax + b, where a and b are real numbers.  We have also talked about the slope of a line and many of its properties. In this post, we will discuss the generalization of the equation of a line in the coordinate plane based on its slope and y-intercept.

We have learned that to get a slope of a line, we only need two points.  We have also learned that given two points on a line, its slope is described as the rise (difference in the y-coordinates) over the run (difference in x-coordinates).  Therefore, if we have two points with coordinates (x_1,y_1) and (x_2,y_2), the slope m is  defined the formula

m = \displaystyle\frac{y_2 - y_1}{x_2 - x_1}.

All the points on a vertical line have similar x-coordinates; therefore, the run {x_2 - x_1} is equal to 0 making m undefined.  From here, we can conclude a vertical line has no slope. » Read more

GeoGebra Tutorial 23 – Spreadsheet, Barchart and Histogram

Note: Although this is the 23rd of the GeoGebra Tutorial Series, just like other GeoGebra tutorials, it is independent from others.  This means that you can follow this tutorial step-by-step, without needing to learn the previous twenty two tutorials.

In this tutorial, we are going to use the spreadsheet to perform basic data representation.  We will plot a frequency table of  grouped data and perform basic computations using the spreadsheet window.  After the data is completed, we will construct a histogram. » Read more

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