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n 1 2 3 4 5
Triangle numbers 1 3 6 10 15
First-order differences 2 3 4 5
Second-order differences 1 1 1 1
The first-order differences are not the same, so the formula for the sequence is not linear. The second-order differences are the same and equal to 1, therefore the formula for the sequence is quadratic. The quadratic formula, in general terms, is T = an2 + bn + c, where T is the value of the triangle number and a, b, and c are constants which need to be found. The value of the constant a is always equal to half the value of the second-order difference.
In this example the value of the second-order difference is 1, therefore a = 1/2. The formula now becomes T = |w2 + bn + c. The values of b and c are found by substituting two pairs of values for T and n into the formula T = ~n2 + bn + c. For instance, when n = 1, T = 1, and when n = 2, T = 3. We write
l = Axl2T&xlTc Substituting T = 1 and ft = 1 into the quadratic
b + c = 0.5 Simplifying the equation
Similarly, by substituting T = 3 and n = 2 into the equation T = ~n2 + bn + c we obtain the simplified equation 2b + c = 1.
We now have two simultaneous equations:
b + c = 0.5 2b + c = 1
Subtracting the first equation from the second equation gives b = 0.5. Substituting b = 0.5 into the first equation 1 gives c = 0.
The formula for the sequence of triangle numbers is T = |w2 + |w. This can be factorized to T = |w(w + 1).
References: Formula, Linear Equation, Patterns, Quadratic Equations, Sequence, Simultaneous Equations, Triangle Numbers.
A digit is any one of the numerals 0,1, 2, 3,4, 5, 6,7, 8, 9. For example, the number 35 is made up of two digits, 3 and 5.
A dihedral angle is the angle between two planes that intersect in a straight line. Suppose two planes intersect in a straight line AS (see figure) and point P is any point on the line AB. A straight line PX is drawn in one plane so that it makes a right angle with the line AS. Similarly, a line PY is drawn in the other plane so that it makes a right angle with the line AS. The dihedral angle is the angle XPY.
References: Angle between Two Planes, Plane.
This is another name for enlargement.
References: Congruent Figures, Indirect Transformation.
A function is discontinuous if its graph is broken into two or more parts that are not connected. If you were drawing a graph of a function, at a point of discontinuity you would have to take your pencil off the paper to continue the drawing of the graph. The function
is discontinuous at the point where x = 1 (see figure a). At this point its graph is broken into two parts. The point of discontinuity is indicated on the graph by drawing a small circle to highlight the fact that there is a gap in the line at the point (1, 2).
y = r
is discontinuous at x = 1 (see figure b).
A function is continuous if its graph has no break in it.
References: Function, Graphs.
Reference: Conversion Factor.
Reference: Frequency Distribution.
Before displacement can be adequately described it is better to say what distance is, because displacement and distance are similar. Distance is the measure of the change in position of an object or point and can take place in any direction. It is the length of the straight line that can be drawn from the starting point (or position) to the final point (position). The usual units of distance are millimeters (mm), centimeters (cm), meters (m), and kilometers (km).
Displacement is the shift from the starting position of an object to its final position, provided that the distance and the direction are given. Its units are the same as the units of distance, provided that the direction is also stated.
Suppose John walks a distance of 3 km from home, which is at point H in the figure. He could finish up at various places, such as work (W), the dairy (D), or the park (P). All these places are a distance of 3 km from home and lie on a circle of radius 3 km with its center at. H. If his wife said he must walk from home 3 km due east so that John finished up at the park, then that instruction would describe a displacement, because the distance and the direction are given. When a displacement is given as an instruction there is only one destination, in this case the park.
DISTRIBUTIVE LAW 161
This is a term used in statistics. When we collect data, which may be a set of observations or a set of measurements, record the frequency of each item of data, and arrange the data in some readable form, it is called a distribution. The distribution of the data may be displayed graphically.
Example. Thirty families in a village were randomly selected and the data collected from them was regarding the number of children in each family. The distribution of family sizes looked like this: