Properties of AP:
If a, b, c, d,…. are in AP and ‘k’ is a constant then
- a – k, b – k, c – k,… will also be in AP
- ak, bk, ck,… will also be in AP
- a/k, b/k, c/k will also be in AP
Geometric progression (GP)
- If in a succession of numbers, the ratio of any term and the previous term is constant then the numbers are said to be in geometric progression.
Properties of GP:
If a, b, c, d,…. are in GP and ‘k’ is a constant, then
- ak, bk, ck,… will also be in GP
- a/k, b/k, c/k,... will also be in GP
Harmonic progression (HP):
- If a, b, c, d,... are unequal numbers then they are said to be in HP if 1/a, 1/b, 1/c,... are in AP
- The ‘nth’ term in HP is 1/(nth term in AP)
Properties of HP:
If a, b, c, d,… are in HP, then
- a + d > b + c
- ad > bc
Standard series:
Arithmetic mean:
The final series is a, a + d, a + 2d,….
Geometric mean (GM):
Harmonic mean:
Relationship between AM, GM, and HM for two numbers a and b:
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