Concept To Remember

Arithmetic Geometric Harmonic Progressions

Concepts of Arithmetic Progressions (AP):

Common difference (d): The common difference in an AP is the constant value by which each term differs from the previous one.
nth term (Tn = a + (n - 1)d): The nth term of an AP can be calculated using the formula Tn = a + (n - 1)d, where ‘a’ is the first term and ’n’ is the number of the term you want to find.
Sum of n terms (Sn = n/2(a + an)) The sum of n terms of an AP can be calculated using the formula Sn = n/2(a + an), where ‘a’ is the first term, ‘an’ is the nth term, and ’n’ is the number of terms.

Concepts of Geometric Progressions (GP):

Common ratio (r): The common ratio in a GP is the constant value by which each term is multiplied to obtain the next term.
nth term (Tn = ar^(n-1)): The nth term of a GP can be calculated using the formula Tn = ar^(n-1), where ‘a’ is the first term, ’n’ is the number of the term, and ‘r’ is the common ratio.
Sum of n terms (Sn = a(rn - 1)/(r - 1)): The sum of n terms of a GP can be calculated using the formula Sn = a(rn - 1)/(r - 1), where ‘a’ is the first term, ‘r’ is the common ratio, and ’n’ is the number of terms.

Concepts of Harmonic Progressions (HP):

Harmonic mean (H = n/(1/a1 + 1/a2 + …)): The harmonic mean of a set of numbers is defined as the reciprocal of the average of their reciprocals.
nth term (Tn = n/(a1 + a2 + … + an)): The nth term of an HP can be calculated using the formula Tn = n/(a1 + a2 + … + an), where ‘a1’, ‘a2’, …, ‘an’ are the terms of the HP and ’n’ is the number of the term you want to find.
Sum of n terms (Sn = n/H): The sum of n terms of an HP can be calculated using the formula Sn = n/H, where ‘H’ is the harmonic mean and ’n’ is the number of terms.

Other Important Concepts:

Properties of arithmetic, geometric, and harmonic means: The arithmetic mean (average) is always greater than or equal to the geometric mean, which in turn is greater than or equal to the harmonic mean.
Arithmetic-Geometric (A.G.) mean: The A.G. mean of two positive numbers is defined as the positive root of their product.
Geometric-Harmonic (G.H.) mean: The G.H. mean of two positive numbers is defined as the harmonic mean of their geometric mean and arithmetic mean.
Sum to infinity of a GP: The sum to infinity of a convergent GP with common ratio r (|r| < 1) is given by the formula S∞ = a/(1 - r), where ‘a’ is the first term and ‘r’ is the common ratio.