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 a_{1}, a_{2}, …….., a_{n}, ………

Let a_{1}, a_{2}, …….., a_{n}, …….. be a sequence. Then the expression a_{1} + a_{2} + ……. + a_{n} + …….. is called the series associated with the given sequence.

II. Arithmetic Progression (AP)

A sequence a_{1}, a_{2}, ……, a_{n}, …….. is called an arithmetic sequence or arithmetic progression if a_{n+1} = a_{2} + d, n ∈ N, where a_{1} 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} = \(\frac{n}{2}\) [t_{1} + t_{n}]

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

III. Geometric Progression (GP):

A sequence a_{1} + a_{2} + ……… + a_{n} + …….. is called Geometric sequence or Geometric progression if \(\frac{a_{k+1}}{a_{k}}=r\), k ≥ 1, where a_{1} 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, ar^{2},…… 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 + ar^{2} + ar^{3} + …… ∞ 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