Let us consider the following examples: Assume that there is a generation gap of 30 years, we are asked to find the number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person might have over 300 years.
Here, the total number of generations = `300 /30 =10`
The number of person’s ancestors for the first, second, third, …, tenth generations are 2, 4, 8, 16, 32, …, 1024. These numbers form what we call a sequence.
The `n^(th)` term is the number at the nth position of the sequence tand is denoted by `a^n`.The `n^(th)` term is also called the general term of the sequence.
A sequence containing finite number of terms is called a finite sequence. For example, sequence of ancestors is a finite sequence since it contains 10 terms (a fixed number).
A sequence is called infinite, if it is not a finite sequence. For example, the sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends.
a sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it. Sometimes, we use the functional notation a(n) for `a_n`.
The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2 , n > 2.
Find `a_(n+1)/a_n`, for n = 1, 2, 3, 4, 5
If S1, S2, S3 are the sum of first n natural numbers, their squares and their cubes, respectively, show that `9S_2^2 = S_3(1 + 8S_1)`