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| Discrete Algebra |
 Sigma-Notations Arithmetic Progressions Geometric Progressions Binomial Theorem Pascal's Triangle
1 + 2 + ... + n 1² + 2² + ... + n² 1³ + 2³ + ... + n³
Basic Conceptions Constituents of the Induction Principle Model Examples Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 |
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An arithmetic progression is a sequence in which each term (after the first) is determined by adding a constant to the preceding term. This constant is said to be the common difference of the arithmetic progression. The following equations express this sentence mathematically: an+1
= an
+ d,
an = a1 + (n – 1) d, where a1 is the first term of the arithmetic progression; an is its nth term; d is the common difference of the arithmetic progression; n is a natural number. Consider the sum of the first n terms of an arithmetic progression:
Let us note that sum (3) consists of equal pairs: a1
+ an
= (a1 + d)
+ (an
– d) = Therefore, the sum Sn holds its value if each its term is replaced by (a1 + an) / 2. Since the sum contains n terms then
In view of the above equality this formula can be also written as Conclusion: Any arithmetic progression is completely determined by any two its quantities.
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