Progression |numeracy-10

Mast Ram MeenaMast Ram Meena गुरुवार, जनवरी 24, 2013
Progression

Arithmetic Progression

An arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
For example, the sequence 3, 5, 7, 9, 11,... is an arithmetic progression with common difference 2.

Arithmetic progression property:
a1 + an = a2 + an-1 = ... = ak+an-k+1

Formulae for the n-th term can be defined as:
an = 1/2 x (an-1 + an+1)

If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the n-th term of the sequence is given by
an = a1 + (n - 1)d, n = 1, 2, ...

The sum S of the first n values of a finite sequence is given by the formula:
S = 1/2(a1 + an)n, where a1 is the first term and an the last.

or
S = 1/2(2a1 + d(n-1))n

 

Example 1: Find the sum of the first 10 numbers from this arithmetic progression 1, 11, 21, 31...
Solution: we can use this formula S = 1/2(2a1 + d(n-1))n
              S = 1/2(2.1 + 10(10-1))10 = 5(2 + 90) = 5.92 = 460

Example 2: The sum of the three numbers in A.P is 21 and the product of their extremes is 45. Find the numbers.

Solution: Let the numbers are be a - d, a, a + d
Then a - d + a + a + d = 21
3a = 21
a = 7
and (a - d)(a + d) = 45
a2 - d2 = 45
d2 = 4
d = +2
Hence, the numbers are 5, 7 and 9 when d = 2 and 9, 7 and 5 when d = -2. In both the cases numbers are the same.

Geometric Progression

A geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.

a geometric sequence can be written as:

aq0=a, aq1=aq, aq2, q3, ... where q ≠ 0, q is the common ratio and a is a scale factor.

Formulae for the n-th term can be defined as:

an = an-1.q
an = a1.qn-1

The common ratio then is:

q =
ak
ak-1

A sequence with a common ratio of 2 and a scale factor of 1 is 1, 2, 4, 8, 16, 32...

A sequence with a common ratio of -1 and a scale factor of 3 is 5, -5, 5, -5, 5, -5,...

If the common ratio is:

  • Negative, the results will alternate between positive and negative.
  • Greater than 1, there will be exponential growth towards infinity (positive).
  • Less than -1, there will be exponential growth towards infinity (positive and negative).
  • Between 1 and -1, there will be exponential decay towards zero.
  • Zero, the results will remain at zero

Geometric Progression Properties

a2k = ak-1.ak+1
a1.an = a2.an-1 =...= ak.an-k+1

Formula for the sum of the first n numbers of geometric progression

Sn = a1 - anq 1  -  q  = a1. 1 - qn 1 - q

Infinite geometric series where |q| < 1

 

If |q| < 1 then an -> 0, when n -> ∞ So the sum S of such a infinite geometric progression is:

S =
1
1 - x
which is valid only for |x| < 1

Harmonic Progressions

A sequence of numbers is said to form a harmonic progression if their reciprocals form an arithmetic progression.
e.g. 1/4, 1/9, 1/14, 1/19 are in H.P. since 4,9, 14, 19 are in A.P.
In general, the numbers 1/a, 1/(a+d), 1/(a+2d), ..., 1/(a+(n+1)d) are in H.P.

Note:

i) The series formed by the reciprocals of the terms of a geometric series is also a geometric series.

ii) There is no general method of finding the sum of a harmonic progression.
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