As per JEE syllabus, the main concepts under sequences are arithmetic progression, geometric progression, harmonic progression, AM-GM-HM inequality and its applications.
Arithmetic Progression (AP)
- General term: tn=a+(n−1)d
- Sum of the terms: Sn=2n(2a+(n−1)d)
- Arithmetic mean and its property: (AM of any n numbers a1,a2,…,an)=na1+a2+⋯+an
- Insertion of arithmetic means between two numbers:
If a and b are two given numbers and a,A1,A2,…,An,b are in AP, then A1,A2,…,An are n AM's between a and b: A1=a+n+1b−a,A2=a+n+12(b−a),…,An=a+n+1n(b−a).
Geometric Progression (GP)
- General term: tn=arn−1
- Sum of the terms: Sn−r−1a(rn−1)
- Geometric mean and its property: (GM of any n numbers a1,a2,…,an)=(a1⋅a2⋅⋅⋅an)n1
- Insertion of geometric means between two numbers:
If a and b are two given numbers and a,G1,G2,…,Gn,b are in GP, then G1,G2,…,Gn are n GM's between a and b: G1=a(ab)n+11,G2=a(ab)n+12,…,Gn=a(ab)n+1n.
Harmonic Progression (H.P.)
- General term: an1=a1+(n−1)d
- Harmonic mean and its property:
If HM of any n numbers a1,a2,…,an is H, then H=a11+a21+⋯+an1n.
- Insertion of harmonic means between two numbers:
If a and b are two given numbers and a,H1,H2,…,Hn,b are in HP, then H1,H2,…,Hn are n HM's between a and b: H11=a1+n+1(b1−a1),H21=a1+n+12(b1−a1),…,Hn1=a1+n+1n(b1−a1).
AM-GM-HM inequality and its applications
- If a1,a2,…,an are all positive real numbers, then
na1+a2+...+an≥(a1⋅a2⋅⋅⋅an)n1≥a11+a21+...+an1n.