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The sum of the first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum of n 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 a1,a2,a3,....,an,... is a sequence, then the expression a1+a2+a3+....+an+... 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., an+1an=constant for all nN

(ii) The constant difference d is called the common difference.

(iii) A sequence is an arithmetic progression if and only if its nth term is a linear expression in n and in such a case the common difference is equal to the coefficient of n.

(iv) If a is the first term and d is the common difference of an A.P., then its nth term is given by an=a+n1d

(v) If an A.P. consists of m terms, then nth term from the end is equal to mn+1th term from the beginning.

(vi) The sum Sn of n terms of an A.P. with first term a and the common difference d is given by Sn=n22a+(n-1)d.

(vii) Sn=n2(a+l), where l=last term=a+n-ld.

(viii) Three numbers a,b,c are in A.P. if 2b=a+c. In such case b is called the arithmetic mean of a and c.

5. Arithmetic mean:

(i) The arithmetic mean of a and b is a+b2.

(ii) If n numbers A1,A2,....,An are inserted between two given numbers a and b such that a,A1,A2,....,An,b is an arithmetic progression, then A1,A2,....,An are known as n arithmetic means between a and b and the common difference of the A.P. is d=b-an+1

(iii) A1+A2+....+An=na+b2

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., an+1an=constant for all nN .

(ii) The constant ratio is called the common ratio of the G.P.

(iii) The nth term of a G.P. with first term a and common ratio r is given by an=arn1.

(iv) If a G.P. consists of m terms, then nth term from the end is mn+1th term from the beginning and is given by armn.

(v) Sum of n terms of a G.P with first term a and common ratio is given by Sn=arn-1r-1 or Sn=a1-rn1-r ; r 1

(vi) Sn=a-lr1-r or Sn=lr-ar-1 (where l 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 a,b,c are in G.P. if b2=ac. If a,b,c are in G.P., then b is known as the geometric mean of a and c.

(ii) Let a and b be two given numbers. If n numbers G1,G2,G3,....,Gn are inserted between a and b such that the sequence a,G1,G2,....,Gn,b is a G.P., then the numbers G1,G2,G3,....,Gn are known as n geometric means between a and b. The common ratio of the G.P. is given by r=ba1n+1

(iii) The geometric mean of a and b is given by ab 

10Relation between arithmetic mean and geometric mean:

If A and G are respectively arithmetic and geometric means between two positive numbers a and b, then

(i) A>G

(ii) The quadratic equation having a, b as its roots is x22Ax+G2=0

(iii) a:b=A+ A2-G2:A-A2-G2

(iv) If AM and GM between two numbers are in the ratio m:n, then the numbers are in the ratio m+m2-n2:m-m2-n2

11. Some standard results of series:

(i) k=1nk=1+2+3+.....+n=nn+12

(ii) k=1nk2=12+22+32+.....+n2=nn+12n+16

(iii) k=1nk3=13+23+33+.....+n3=nn+122