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.

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Example

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:

$$a_{n}=a_{1}+(n-1)d$$

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:

$$S_{n}=\frac{n}{2}(a_{1}+a_{n})$$

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Example

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:

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

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Video lesson

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