Question

I want all formulas related to sequence and series

That is related to Harmonic progression , AritmArith progression , Geometric progression and special progression

Answer 1

• nth term of an AP = a + (n-1) d
• Arithmetic Mean = Sum of all terms in the AP / Number of terms in the AP
• Sum of ‘n’ terms of an AP = 0.5 n (first term + last term) = 0.5 n [ 2a + (n-1) d]

• If ‘a’ is the first term and ‘r’ is the common ratio,
• nth term of a GP = a r^n-1
• Geometric Mean = nth root of product of n terms in the GP
• Sum of ‘n’ terms of a GP (r < 1) = [a (1 – rⁿ)] / [1 – r]
• Sum of ‘n’ terms of a GP (r > 1) = [a (rⁿ– 1)] / [r – 1]
• Sum of infinite terms of a GP (r < 1) = (a) / (1 – r)

A sequence of numbers is called a harmonic progression if the reciprocal of the terms are in AP. In simple terms, a,b,c,d,e,f are in HP if 1/a, 1/b, 1/c, 1/d, 1/e, 1/f are in AP.

• For two terms ‘a’ and ‘b’,
• Harmonic Mean = (2 a b) / (a + b)

• For two numbers, if A, G and H are respectively the arithmetic, geometric and harmonic means, then

• A ≥ G ≥ H
• A H = G², i.e., A, G, H are in GP.

Sum of first n natural numbers=n(n+1)/2

Sum of squares of first n natural numbers
=n(n+1)(2n+1)/6


Sum of first n odd natural numbers=n^2

Sum first n even natural numbers=n(n+1)