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Mathematics Magazine for Grades 1-12  

 

12-2004

 

 

 

Grade 11

Theory:

Graphs

Solutions from the Previous Issue: 

1.        Find the tenth term of the arithmetic sequence 2,5,8,..

Solution:

The n-th term of an arithmetic sequence is given by:

an = a1 + (n - 1)∙d

By subtracting any given term from the following term we find that the common difference is d = 3. From the terms given, we know that the first term is a1 = 2. From the statement of the problem the desired term is the tenth, or n = 10

Thus we find that the tenth term is:

a10 = 2 + (10 - 1)∙3 = 29

2.        Find the common difference between successive terms of the arithmetic sequence for which the first term is 5 and the 32nd term is 119.

Solution:

The n-th term of an arithmetic sequence is given by:

an = a1 + (n - 1)∙d                           (1)

We are to find d given that a1 = 5 and a32 = - 119, and n = 32. Substitution in (1) gives:

-119 = 5 + (32 1) d

d = - 4  

3.        How many numbers between 10 and 1000 are divisible by 6?

Solution:

We must first find the smallest and tle largest numbers in this range that are divisible by 6. These numbers are 12 and 996. The common difference between one multiple of 6 and the next is 6. Thus we can solve this as and arithmetic sequence an = a1 + (n - 1)∙d                (1)

with a1 = 12 and an = 966 and d = 6.

Substituting in equation (1) we have:

996 = 12 + (n - 1) ∙ 6;

996 = 12 + 6n - 6;

996 = 6 + 6n;

990 = 6n

n =165

4.        Find the sum of the first 1000 positive integers.

Solution:

The sum of the first n terms of an arithmetic sequence is:

a1000 = 1000; n = 1000; d = 1

= 500500

Proposed Problems:

1.        The cost of a long-distance phone call from Toronto to New York City is 69 cents for the first minute and 58 cents for each additional minute (or part of a minute). Draw the graph of the cost C (in dollars) of the phone call as a function of the time t ( in minutes).

2.        If  and find the function f + g, f g

3.        If f(x) = x2 and g(x) =x 3 find the composite functions f  o g and g o f and their domains

4.      If  and find the function  F= f  o g

 

 
 
 

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