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This essential trains for: SAT-I, AMC-8, AMC-10, SAT-II, GMAT, Math Kangaroo 7-8, Math Kangaroo 9-10.

'Arithmetic progression' is another term for 'arithmetic sequence'. In fact, it is the more traditional terminology.

Definitions

If each term of a sequence differs from the previous (and from the next) term by a fixed amount, we call the sequence arithmetic.

The fixed amount by which adjacent terms differ is called the common difference.

We can build the entire sequence if we know the first term of the sequence and the common difference.

The N-th term of the arithmetic sequence:

Let a1 be the first term of the sequence and d the common difference.

 Rank of term Name of term Term value

It is easy to see from the table that the N-th term would be:

The sum of the arithmetic sequence

We want to compute the sum of the first N terms. Notice how the indexing of the terms goes from 0 to N-1.

Let us write vertically the list of sums of 1,2,3 terms and more up to N:

Add all these equalities: all the left sides together and all the right sides together:

Notice how many of the terms in the sums cancel out, leaving:

We can now use the formula:

to obtain the final result:

The sum of the sequence if it begins with at an ordinal larger than 1:

If you have to add only the terms between the p-th and the q-th, then you simply subtract the sum from 1 to p from the sum from 1 to q:

Useful facts: Given a sequence of three consecutive terms of an arithmetic sequence, the middle term is the arithmetic mean (average) of its neighbors.

Denote the middle term with m, then the three consecutive terms look like this:

Obviously, the middle term is the average: