
The sum of the first three terms of a G.P. is and the sum of the next three terms is Determine the first term, the common ratio and the sum of terms of the G.P.
Important Points to Remember in Chapter -1 - Sequences and Series from NCERT Mathematics Textbook for Class 11 Solutions
1. Sequence:
A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule.
(i) A sequence containing a finite number of terms is called a finite sequence.
(ii) A sequence is called an infinite sequence if it is not a finite sequence.
2. Series:
lf is a sequence, then the expression is called a series.
3. Progressions:
Those sequences whose terms follow certain patterns are called progressions.
4. Arithmetic progression:
(i) A sequence is called an arithmetic progression if the difference of a term and the previous term is always same, i.e., constant for all .
(ii) The constant difference is called the common difference.
(iii) A sequence is an arithmetic progression if and only if its term is a linear expression in and in such a case the common difference is equal to the coefficient of
(iv) If is the first term and is the common difference of an A.P., then its term is given by
(v) If an A.P. consists of terms, then term from the end is equal to term from the beginning.
(vi) The sum of terms of an A.P. with first term and the common difference is given by .
(vii) , where last term
(viii) Three numbers are in A.P. if In such case is called the arithmetic mean of and
5. Arithmetic mean:
(i) The arithmetic mean of and is .
(ii) If numbers are inserted between two given numbers and such that is an arithmetic progression, then are known as arithmetic means between and and the common difference of the A.P. is
(iii)
6. Properties of arithmetic progressions:
(i) If a constant is added or subtracted to each term of an A.P, then the resulting terms of the sequence are also in A.P with the same common difference.
(ii) If each term of an A.P is multiplied or divided by a non-zero constant, then the resulting sequence also forms an A.P.
7. Geometric progression:
(i) A sequence is called an geometric progression if the ratio of a term and the previous term is always same, i.e., constant for all .
(ii) The constant ratio is called the common ratio of the G.P.
(iii) The term of a G.P. with first term and common ratio is given by .
(iv) If a G.P. consists of terms, then th term from the end is term from the beginning and is given by .
(v) Sum of terms of a G.P with first term and common ratio is given by or
(vi) or (where is the last term)
8. Properties of geometric progression:
(i) If all the terms of G.P. be multiplied or divided by the same non-zero constant, then it remains a G.P. with the same common ratio.
(ii) The reciprocals of the terms of a given G.P. form a G.P.
(iii) If each term of a G.P. be raised to the same power the resulting sequence also forms a G.P.
9. Geometric mean:
(i) Three numbers are in G.P. if If are in G.P., then is known as the geometric mean of and
(ii) Let and be two given numbers. If numbers are inserted between and such that the sequence is a G.P., then the numbers are known as geometric means between and The common ratio of the G.P. is given by
(iii) The geometric mean of and is given by
10. Relation between arithmetic mean and geometric mean:
If and are respectively arithmetic and geometric means between two positive numbers and , then
(i)
(ii) The quadratic equation having as its roots is
(iii)
(iv) If AM and GM between two numbers are in the ratio then the numbers are in the ratio
11. Some standard results of series:
(i)
(ii)
(iii)