Progresions 2
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PROGRESSIONS (2)
HARMONIC PROGRESSION
A series is said to be in harmonic progression if the series consisting of the reciprocals of each term in the series is in AP. In other words a series a1, a2, a3, a4,………, an is in HP if 1/ a1,1/a2, 1/a3, 1/a4,………, 1/an are in AP.
The nth term of an harmonic progression is denoted by tn and is given by
| tn = 1/[a + (n-1)d] |
HARMONIC MEAN
If two numbers a and b be in AP then the harmonic mean between them is given as
| H = 2ab/(a+b) |
NOTE: the numbers a, H and b are in HP
n- HARMONIC MEANS
The numbers H1, H2, H3, H4,………, Hn are said to be harmonic means between the numbers a and b if the series
a, H1, H2, H3, H4,………, Hn ,b
is a HP. The nth mean Hn is given as
| Hn = [(n+1)ab]/n(a + b) |
- SUM OF N NATURAL NUMBERS
The sum of n natural numbers is given by
S = 0.5n(n+1)
- SUM OF N NATURAL SQUARES
The sum of n natural squares is given by
S = (1/6)n(n+1)(2n+1)
- SUM OF N NATURAL CUBES
The sum of n natural cubes is given by
S = (1/4)n2(n+1)2