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BICTE First Semester Mathematics - Sequences and Series Pdf Download
Sequence: A set of numbers arranged according to definite rule is called sequence.
e.g. i) 2, 5, 8, 11, 14
ii) 4, 8, 16, 32, 64
Series: Sum of terms of sequence is called series.
e.g. 2+5+8+11+14
Finite sequence: A sequence containing finite number of terms is called finite sequence.
e.g. 2, 5, 8, 11, 14
Infinite sequence: A sequence containing infinite number of terms is called infinite sequence.
e.g. 2, 5, 8, 11, 14, ....
Types of sequence:
- Arithmetic sequence (AP)
- Geometric sequence (GP)
- Harmonic Sequence (HP)
- Arithmetic sequence (AP)
It is the sequence having common difference(d)
e.g. 2, 5, 8, 11
\(d=t_2-t_1=3\)
\(d=t_3-t_2=3\)
nth term of AP\((t_n)=a+(n-1)d\)
sum of n terms\((S_n)=\frac{n}{2}(a+b)\)
Also, \(S_n=\frac{n}{2}(2a+(n-1)d) \)
where, a=first term, b= last term and n=no. of terms - Geometric sequence (GP)
It is the sequence having common ratio.
e.g. 4, 8, 16, 32, 64
\(r=\frac{t_2}{t_1}=\frac{8}{4}\)
\(r=\frac{t_3}{t_2}=\frac{16}{8}\)
nth term of GP\((t_n)=ar^{n-1}\)
sum of n terms\((S_n)=\frac{a(r^n-1)}{r-1}\)
where, a=first term, r=common ration and n=no. of terms - Harmonic Sequence (HP)
The reciprocal of AP is HP.
e.g.\( \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}\)
If the reciprocal of given sequence is AP then it is HP. Reciprocal of above sequence is 3, 5, 7, 9 which is in AP.
Mean:
- Arithmetic mean (A.M.)
A.M. = \( \frac{a+b}{2}\) - Geometric mean (G.M.)
G.M. = \(\sqrt{ab}\) - Harmonic mean (H.M.)
H.M. = \(\frac{2ab}{a+b}\)
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