# Progressions: Other important concepts

## Important Concepts

Sum of first ‘n’ natural numbers

Sum of the squares of the first ‘n’ natural numbers

Sum of the cubes of the first ‘n’ natural numbers

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.

The 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.

1. If every term of an A.P. is multiplied (or divided) by a non-zero constant, then the resulting sequence is also an A.P.

The Sum of the terms of the new series = constant × sum of the terms of the old series

1. 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.
2. The product of two different G.P. is also a G.P. with the common ratio equal to the product of the common ratios of the original progressions.

## Relation between AM, GM, and HM.

For any set of numbers, the arithmetic mean is always greater than or equal to the geometric mean which is always greater than or equal to the harmonic mean. i.e. A.M. ≥ G.M. ≥ H.M. Also,

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