Plus One Maths Notes Chapter 9 Sequences and Series

Kerala State Board New Syllabus Plus One Maths Notes Chapter 9 Sequences and Series.

Kerala Plus One Maths Notes Chapter 9 Sequences and Series

I. Sequence and Series
A sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it of the type {1, 2, 3, ….., k}.
Generally denoted by a₁, a₂, …….., a_n, ………
Let a₁, a₂, …….., a_n, …….. be a sequence. Then the expression a₁ + a₂ + ……. + a_n + …….. is called the series associated with the given sequence.

II. Arithmetic Progression (AP)
A sequence a₁, a₂, ……, a_n, …….. is called an arithmetic sequence or arithmetic progression if a_n+1 = a₂ + d, n ∈ N, where a₁ is called the first term and the constant term d is called the common difference of the AP.

Standard form of an AP:
a, a + d, a + 2d, …….. where a is the first term and d is a common difference.

If a constant is added to each term of an AP, the resulting sequence is also an AP.

If a constant is subtracted to each term of an AP, the resulting sequence is also an AP.

If each term of an AP is multiplied by a constant k, the resulting sequence is also an AP. But the resulting AP will have a common difference kd.

If each term of an AP is divided by a constant k, the resulting sequence is also an AP. But the resulting AP will have a common difference \(\frac{d}{k}\).

nth term, a_n = a + (n – 1)d

Sum of n terms, S_n = \(\frac{n}{2}\) [2a + (n – 1)d]

S_n = \(\frac{n}{2}\) [t₁ + t_n]

Arithmetic mean between a and b is \(\frac{a+b}{2}\)

III. Geometric Progression (GP):
A sequence a₁ + a₂ + ……… + a_n + …….. is called Geometric sequence or Geometric progression if \(\frac{a_{k+1}}{a_{k}}=r\), k ≥ 1, where a₁ is called the first term and the constant term r is called the common ratio of the AP.

Standard form of a GP:
a, ar, ar²,…… where a is the first term and r is a common difference.

nth term, t_n = ar^n-1

Sum of n terms,

Geometric mean between a and b is √ab

Arithmetic mean ≥ Geometric mean.

Infinite G.P, and its Sum G. P. of the form a + ar + ar² + ar³ + …… ∞ is called infinite G.P.
S_∞ = \(\frac{a}{1-r}\); |r| < 1

‘Infinite Series Calculator‘ is an online tool that helps to calculate the summation of infinite series for a given function.

IV. Special Series