#MathsInMinutes Day 45: Arithmetic progression
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#MathsInMinutes Day 45: Arithmetic progression
An arithmetic progression is an ordered list of numbers in which the difference between successive terms is a constant. An example is 0, 13, 26, 39, 52, …, where the constant common difference is 13. If this common difference is positive, a sequence like this will tend to infinity. If the common term is negative, the sequence tends to negative infinity. The recently proved Green-Tao theorem describes the prevalence of long arithmetic progressions of prime numbers.
Partial sums of an arithmetic progression are relatively simple to calculate using a little trick. For instance, what is the sum of the numbers 1 to 100? The easy way to do this is to list the sum twice, once forwards and once backwards, making columns that sum to 101. Since there are 100 of these, the total sum is 100 multiplied by 101, divided by 2. In general, this argument shows that the sum of any arithmetic progression is given by:
a + 2a + 3a + … + na = 0.5a(n+1)
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