Use of gradient and area in arithmetic sequence and series

 

Gradient

An arithmetic sequence is characterized by having a common difference, d.

 

Let  T(1) = a  be the first term, d be the common difference and T(n) be the general term of an arithmetic sequence.

 

Then d   = T(2) – T(1) = T(3) – T(2)

              = …. = T(n) – T(n-1)

 

We like to draw as in the diagram 

        T(1) = a, T(2) = a +d , …., T(n).   

 

The coordinates of the top points of vertical lines are (1, a), (2, a+d), ….. , (n,T(n)).

By considering the gradient of the first two points,

.

 

Since the given sequence is arithmetic, the red line is straight with gradient d.

 

If we find the gradient of the line using the first and the last point, (1, a)  and  (n, T(n)),   

we get

 

 

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Example 1

In the arithmetic sequence  9, 1, -15, ….  , find the 20th term.

 

Solution

Let A(1,9), B(2,1) and C(20, T(20)).

The three points A, B, C are collinear.

\     T(20) = - 143.

 

 

Example 2

  Insert 6 arithmetic means between the numbers  -18 and 3.

 

Solution

  Consider the points A(1, -18) and B(8, 3).

 

\ The 6 arithmetic means are –15, -12, -9, -6, -3, 0.

 

 

Example 3

 

  In the arithmetic sequence the fourth term is 10, the seventh term is 19.

Find the common difference d and the 20th term

 

Solution

  Let A(4, 10), B(7, 19), C(20, T(20))

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  \ T(20) = 58.

 

 

Example 4

 

  In the arithmetic sequence  T(p) = q and T(q) = p, where p ≠ q. Find T(p+q).

 
Solution

  Let A(p, q), B(q,p), C(p+q, T(p+q)).

  A, B and C are collinear.

        \  and we have

        Solving, we get   T(p+q) = 0.

 

 

Area

 

We can use “area” formulae for calculating the numbers of dots in the diagrams:

 

(1)          Applying the “area” of rectangle:

 

Number of dots = 3 x 4 = 12

 

 

 

 


(2)          Applying the “area” of trapezium,

 

Number of dots

  =

  = 16

 


 


        As we can see, the formulae of area of triangle, rectangle, trapezium etc can be used in counting the number of dots if the dots are arranged in a lattice pattern.

 

        Let S(n) = T(1) + T(2) + … + T(n) be the sum of the terms of the arithmetic sequence and we get an arithmetic series.

   

      Now, study the diagram in the right:

 

The first vertical column of dots represents T(1).

The last vertical column of dots represents T(n).

There are altogether n columns.

 

 

 

 

S(n) is the sum of all columns of dots and we can apply the area formula for trapezium.

 

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