|Framing the rule
Ex 1: 4, 6, 8, 10, . . . .
In above: 4, 6, 8, 10 . . . . are called terms of the sequence.
Clearly, we observe that each term is obtained by adding 2 to the previous term.
In this sequence, the rule is: to get a term, add 2 to the previous term.
Ex 2: 2, 6, 18, 54, . . . .
In above: 2, 6, 18, 54. . . . are called terms of the sequence.
Clearly, we observe that each term is obtained by multiplying the previous term with 3
In this sequence, the rule is: multiply a term by 3 to get the next term.
Ex 3: 12, 6, 0, – 6, – 12 . . . .
In the above sequence, the first term = 12
Second term = 12 – 6 = 6
Third term = 6 – 6 = 0
Fourth term = 0 – 6 = – 6
Clearly, we observe that each term is obtained by subtracting 6 from the previous term.
In this sequence, the rule is: subtract 6 from the previous term.
Ex 4: 4, 9, 16, 25, 36
This is a finite sequence and is equal to
22, 32, 42, 52, 62
In this sequence, the terms are squares of the natural numbers from n = 2 to n = 6.
A set of numbers arranged in a definite order according to some definite rule is called a sequence. Each number of the set is called a term of the sequence.
Finite and Infinite sequence:
A sequence is called finite or infinite according to the number of terms in it is finite or infinite.
General term of sequence:
Let us consider the sequence of "cubes" of natural numbers: 1, 8, 27, 64, . . . .
The different terms of a sequence are usually denoted by t1, t2, t3, . . . etc.
Here, the subscript (always a natural number) denotes the position of the term in the sequence.
Thus, in the above sequence t1 = 1; t2 = 8; t3 = 27 . . . . . etc.
Hence, First term = t1 = 1, Second term = t2 = 8, Third term = t3 = 27, . . . etc.
In general, nth term = tn, which is called general term of the sequence.
Often, it is possible to express the rule which generates the various terms of the sequence in terms of an algebraic formula.
In the above sequence 1, 8, 27, 64, 125 . . ., nth term = tn = n3.
Thus, the rule for the above sequence is n3, where n is any natural number.
If a1, a2, . . ., an is a sequence of numbers,
then the expression a1 + a2 + . . . + an
is called series associated with the given sequence.
Like a sequence, a series may also be finite or infinite.
It is common to represent a series compactly using the Σ (sigma) symbol. Sigma indicates a summation as:
ai = a1 + a2 + a3 + . . . + an for a finite series.
Note that 'i' takes the values from '1' to the number indicated at the top of the sigma symbol.
(The L.H.S. is read as "sigma ai, i equal to 1 to n")
ai = a1 + a2 + a3 + . . . + an + . . . ∞ for an infinite series.