HOW TO DETERMINE IF A SEQUENCE IS ARITHMETIC
If a sequence is arithmetic, the difference between any two consecutive terms will be same along the sequence.
Let a1, a2, a3, a4, a5, a6, ............ be a sequence.
Let 'd' be the difference between the first term and a1 and second term a2. That is,
d = a2 - a1
In the above sequence, if the difference between any two consecutive terms is 'd' along the sequence, then the sequence is arithmetic.
d1 = a2 - a1
d2 = a3 - a2
d3 = a4 - a3
d4 = a5 - a4
d5 = a6 - a5
d1 = d2 = d3 = d4 = d5 = d
The difference 'd' is called common difference.
Determine whether the following sequences are arithmetic. If so, state the common difference.
Example 1 :
7, 6, 5, 4, ...........
Solution :
|
d1 = a2 - a1 = 6 - 7 = -1 |
d2 = a3 - a2 = 5 - 6 = -1 |
d3 = a4 - a3 = 4 - 5 = -1 |
d1 = d2 = d3 = -1
The difference between any two consecutive terms is same along the sequence. So, the given sequence is arithmetic.
Example 2 :
10, 12, 15, 18, ...........
Solution :
|
d1 = a2 - a1 = 12 - 10 = 2 |
d2 = a3 - a2 = 15 - 12 = 3 |
d3 = a4 - a3 = 18 - 15 = 3 |
d1 ≠ d2
The difference between any two consecutive terms is not same along the sequence. So, the given sequence is not arithmetic.
Example 3 :
9, 5, -1, -5, ...........
Solution :
|
d1 = a2 - a1 = 5 - 9 = -4 |
d2 = a3 - a2 = -1 - 5 = -6 |
d3 = a4 - a3 = -5 - (-1) = -5 + 1 = -4 |
d1 ≠ d2
The difference between any two consecutive terms is not same along the sequence. So, the given sequence is not arithmetic.
Example 4 :
-15, -11, -7, -3, ...........
Solution :
|
d1 = a2 - a1 = -11 - (-15) = -11 + 15 = 4 |
d2 = a3 - a2 = -7 - (-11) = -7 + 11 = 4 |
d3 = a4 - a3 = -3 - (-7) = -3 + 7 = 4 |
d1 = d2 = d3 = 4
The difference between any two consecutive terms is same along the sequence. So, the given sequence is arithmetic.
Example 5 :
-0.3, 0.2, 0.7, 1.2, ...........
Solution :
|
d1 = a2 - a1 = 0.2 - (-0.3) = 0.2 + 0.3 = 0.5 |
d2 = a3 - a2 = 0.7 - 0.2 = 0.5 |
d3 = a4 - a3 = 1.2 - 0.7 = 0.5 |
d1 = d2 = d3 = 0.5
The difference between any two consecutive terms is same along the sequence. So, the given sequence is arithmetic.
Example 6 :
2.1, 4.2, 8.4, 17.6, ...........
Solution :
|
d1 = a2 - a1 = 4.2 - 2.1 = 2.1 |
d2 = a3 - a2 = 8.4 - 4.2 = 4.2 |
d3 = a4 - a3 = 17.6 - 8.4 = 9.2 |
d1 ≠ d2
The difference between any two consecutive terms is not same along the sequence. So, the given sequence is not arithmetic.
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