Class 11th mathematics(sequences and series)

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Let us assume that there is a generation gap of 25 years and we are required to find total ancestors that a person might have over 400 years. Here, the total generations = 400/25 = 16. The number of ancestors for the 1st, 2nd,3rd, . . . 16th generations form what we call a sequence. The different numbers occurring in a sequence are known as its terms denoted by m1, m2, m3, . . . . . , mn, . . . . . etc.. [Here, the position of the term is denoted by the subscript].
A sequence m1, m2, m3, . . . . , mn is said to be in arithmetic sequence or progression if mn+1=mn+d, n ∈ N, where the 1st term and the common difference of an A.P are denoted by m1 and d respectively.
Let us consider an Arithmetic Progression with p as the 1st term and d as the common difference, i.e., p, p + d, p + 2d, . . . . Then the pth term of the A.P. is given by pn = p + (n – 1)d. If a constant term is added, subtracted, multiplied, or divided to an arithmetic progression then the resultant sequence is also an Arithmetic Progression. Given two numbers p and q. A number A can be inserted between them such that p, A, q is an AP. Such number is known as the A.M (arithmetic mean) of the numbers p and q. The sum of the 1st n terms of an A.P (Arithmetic Progression) is calculated by
Sn=n2[2a+(n−1)d]
The 1st term of a G.P is denoted by ‘a’ and the common ratio by ‘r’. The general term of a Geometric Progression is given by an=arn−1and the sum of the 1st n terms is given by,
Sn=a(rn−1)r−1ora(1−rn)1−r
The geometric mean of any two +ve numbers p and q is given by G = pq−−√and the sequence p, G, q is also a G.P.

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