Sequences and Series Class 11 Notes- Chapter 9

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 \(m_{n+1}=m_{n}+d\), 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

\(S_{n}=\frac{n}{2}\left [ 2a\;+\;(n-1)d \right ]\)

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 \(a_{n}=ar^{n-1}\) and the sum of the 1st n terms is given by,

\(S_{n}=\frac{a(r^{n}-1)}{r-1}\;\;or\;\;\frac{a(1-r^{n})}{1-r}\)

The geometric mean of any two +ve numbers p and q is given by G = \(\sqrt{pq}\) and the sequence p, G, q is also a G.P.

Sequences and Series Class 11 Practice Problems

  1. If Geometric and Arithmetic Mean of two +ve numbers p and q are 12 and 10 respectively, find both the numbers.
  2. The fourth term of a Geometric Progression (G.P.) is square of its 2nd term and its 1st term is -6. Find its 6th term.
  3. Determine the sum of n terms of the sequence, 7, 77, 777, 7777, . . . . .
  4. Determine the sum of n terms of the series: 3 + 9 + 17 + 27 + 39 + . . . .
  5. Find the 24th term of a sequence defined by pn = (p – 1) (3 + p) (2 – p)?

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Practise This Question

An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is