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Ch. 6. Sequences and Series - 1

Sequence

A sequence is a function whose domain is the set N of natural numbers.

Sequence is denoted by ‘a’ and the nth term in the sequence a(n) is denoted by a_{n/sub>.
A sequence whose range is a subset of R is called a real sequence.
Representation of a sequence
One way is to list its first few terms till the rule for writing down other terms becomes clear.
Another way is to represent a real sequence is to give a rule of writing the nth term of the sequence.
Series
If a1, a2, a3, … is a sequence, then the expression a1+a2+a3+… is a series.
Progressions
It is not necessary that the terms of a sequence always follow a certain pattern or they are described by some explicit formula for the nth term. Those sequences whose terms follow certain patterns are called progressions.
2 Arithmetic progression
3. General term of A.P.
nth term = a +(n-1)d
6.4 Selection of terms in A P.
6.5 Sum to n terms of AP
Sn = ½ n(a + l)
Where
n = number of terms in the progressin
a = first term
l = last term
Sn = ½ n{2a+(n-1)d}
6. Properties of AP
7. Insertion of arithmetic means
When three quantities are in arithmetical progression, the middle one is called the arithmetic mean of the other two.
a-d, a, and a+d are in arithmetic progression. ‘a’ is the arithmetic mean of a-d and a+d.
For model problems on A.P. visit
http://iit-jee-maths-aps.blogspot.com/2008/06/progressions-model-problems-1.html
8. Geometric progression (GP)
9. General term of a GP
nth term = arn-1
10. Selection of terms in a GP
11. sum of n terms of GP
Sn = a(1-rn)/(1-r)
12. Sum of infinite GP
S = a/(1-r) when -1
13. Properties of GP
14. Insertion of Geometric mean between two given numbers
When three quantities are in geometrical progression, the middle one is said to be the geometric mean of the other two.
a/r, a, and ar are in GP. ‘a’ is the geometric mean of a/r and ar.
√(ab) is the geometric mean of a and b
15. Some important properties of AM and GM between two quantities
16. Arithmetic Geometric sequence
17. Sum of n terms of arithmetic geometric sequence
18. Sum to n terms of some special sequences
19. Miscellaneous sequences and series
20. harmonic progressions
21. Properties of arithmetic, geometric and harmonic means between two given numbers.
For model problems on this chapter visit
http://iit-jee-maths-aps.blogspot.com/2008/06/progressions-model-problems-1.html
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