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Mathematics Magazine for Grades 1-12  

4/2004

 

     

You should be more afraid of a stupid man than of an evil one.
 - Christina of Sweden

Grade 11

Theory:

Arithmetic Progression

An arithmetic progression is a sequence in which each term after the first is formed by adding a fixed amount, called the common difference, to the preceding term.

 

If a is the first term, d is the common difference, and n is the number of terms an arithmetic progression, the successive terms are

a, a + d, a + 2d, a + 3d,.a + (n-1)d

Thus, the last term (or nth term) l is given by

l= a + (n - 1)d

The sum S of the n terms of this progression is given by

 or 

Arithmetic Means

The terms between the first and last terms of an arithmetic progression are called arithmetic means between these two terms. Thus to insert k arithmetic means between two numbers is to form an arithmetic progression of (k+2) terms having the two given numbers as the first and the last terms.

Geometric Progression

A geometric progression is a sequence in which each term after the first is formed by multiplying the preceding term by a fixed number, called the common ratio.

If a is the first term, r is the common ratio, and n is the number of terms, the geometric progression is:

a, a∙r, a∙r2, .. a∙rn-1

 

Thus, the last ( or nth ) term l is given by l = a∙rn-1.

The sum S of the first n terms of the geometric progression is given by:  or

Geometric Means

The terms between the first and the last terms of a geometric progression are called geometric means between two terms. Thus, to insert k geometric means between two numbers is to form a geometric progression of k+2 terms having the two given numbers as the first and last terms.

Solutions from the Previous Issue: 

1.        Find the twentieth term and the sum of the first 20 terms of the arithmetic progression 4, 9, 14, 19.

Solution: For this progression a = 4, d = 5, and n = 20; The twentieth term is l = a + (n - 1) ∙ d = 4  + 19 ∙ 5 = 99. And the sum of the first 20 terms is: = 1030

 

2.        Insert five arithmetic means between 4 and 22.

Solution: We have a = 4, l = 22, and n = 5 + 2 = 7. Then 22 = 4 + 6 ∙ d and d = 3. The first mean is 4 + 3 = 7, the second is 7 + 3 = 10, and so on. The required means are: 7, 10, 13, 16, 19 and the resulting progression is 4, 7, 10, 13, 16 19, 22

 

3.        Find the arithmetic mean of the two numbers a and l.

Solution: We seek the middle term of an arithmetic progression of three terms having a and l as first and third terms, respectively. If d is the common difference, then a + d = l d and d = (l - a). The arithmetic mean is a + d = a + (l - a) = (l + a).

 

4.        Find the ninth term and the sum of the first mine terms of the geometric progression 8, 4, 2, 1,.

Solution:  Here a = 8, and r = , and n = 9; The ninth term is

l = arn-1= The sum of the first nine terms is:

Proposed exercises: 

Proposed by Mihai Rosu Professor Toronto

1.        Find vector and parametric equations of the line that passes through the point P and direction vector m.

a)       P(-2,4),          b) P( ,3),

2.        Find a vector and parametric equations of the line :

       a) ,       b) ,

3.        Determine the coordinates of three points corresponding to the parameters 0, 1, and 2 on each line:

a)          

b)

4.        Find vector and parametric equations of the line that passes through the point  and is parallel to: a) the vector ; b) the line .

5.        Show that both lines  and  contain the point . Find the acute angle of intersection of these lines, to the nearest degree.