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# Important Points in Arithmetic, Geometric and Harmonic Progressions

Progression is a sequence of numbers whose terms increase or decrease continuously based on a certain pattern. The pattern can be addition of a constant (Arithmetic progression) or multiplication by a constant (Geometric progression).

### Important Points in Series and Progressions

1. Sum of first n natural numbers 1 + 2 + 3 + …. + n = n (n + 1) / 2
2. Sum of the squares of the first n natural numbers 12 + 22 + ... + n2 = (1/6) n (n + 1)(2n + 1)
3. Sum of the cubes of the first n natural numbers 13 + 23 + 33 + …. + n3 = (n (n + 1))2 / 2
4. If three numbers a, b, c are in arithmetic progression, then the middle term ‘b’ will be the arithmetic mean of the three terms and is given by b = (a + c) / 2
5. If a constant number is added (or subtracted) to each term of a given A.P, then the resulting sequence will also be an A.P, and it will have the same common difference as that of the original A.P.
Sum of the terms of the new series = Sum of the terms of the old series + n × constant (where n is the number of terms in the series).
6. If every term of an A.P. is multiplied by a non-zero constant (or divided by a non-zero fixed constant), then the resulting sequence is also an A.P.
Sum of the terms of the new series = constant × sum of the terms of the old series
7. If every term of a G.P. is multiplied by a non-zero constant (or divided by a non-zero fixed constant), then the resulting sequence is also a G.P. with the same common ratio.
8. The product of two different geometric progressions is also a geometric progression with the common ratio equal to the product of the common ratios of the two original progressions.
9. For any set of numbers, the arithmetic mean is always greater than or equal to geometric mean which is always greater than or equal to the harmonic mean.
AM >= GM >= HM

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