# 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, …….., an, ………
Let a1, a2, …….., an, …….. be a sequence. Then the expression a1 + a2 + ……. + an + …….. is called the series associated with the given sequence.

II. Arithmetic Progression (AP)
A sequence a1, a2, ……, an, …….. is called an arithmetic sequence or arithmetic progression if an+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, an = a + (n – 1)d

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

Sn = $$\frac{n}{2}$$ [t1 + tn]

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

III. Geometric Progression (GP):
A sequence a1 + a2 + ……… + an + …….. 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, tn = arn-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