3.2 Arithmetic progression

Arithmetic sequences and series

An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.

Example 1

2,4,6,8,10….is an arithmetic sequence with the common difference 2.

If the first term of an arithmetic sequence is a₁ and the common difference is d, then the nth term of the sequence is given by:

an=a1+(n−1)d$$a_n=a_1+(n-1)d$$

An arithmetic series is the sum of an arithmetic sequence. We find the sum by adding the first, a₁ and last term, a_n, divide by 2 in order to get the mean of the two values and then multiply by the number of values, n:

Sn=n2(a1+an)$$S_n=\frac{n}{2}(a_1+a_n)$$

Example 2

Find the sum of the following arithmetic series 1,2,3…..99,100

We have a total of 100 values, hence n=100. Our first value is 1 and our last is 100. We plug these values into our formula and get:

S100=1002(1+100)=5050$$S_{100}=\frac{100}{2}(1+100)=5050$$

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Video lesson

Find the 20^th value of the following sequence 1,4,7,10….