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From the experts
Chapter 11 : Progression
A succession of numbers formed and arranged in a definite order
according to a certain definite rule is called a sequence or progression.
The number occurring at the nth place of a sequence is called
its nth term or the general term, to be denoted by tn.
A sequence is said to be finite or infinite according as the number
of distinct terms in it is finite or infinite.
We would learn about three types of progressions: Arithmetic,
Geometric and Harmonic.
Arithmetic progression
It is a sequence in which each term, except the first one, differs
from its preceding term by a constant, called the common difference.
So, for examples 1, 6, 11, 16, …., so on and 1, – 1 , – 3, – 5,
…., are A.P.
In the examples above, the difference between any two successive
numbers is equal to 5 and – 2 respectively. This difference is
called the Common Difference.
The general form of expressing this series is a, a + d, a + 2d,
a + 3d, … so on.
The standard notations are as follows.
a = The first term, d = Common difference, Tn = The nth term
l = The last term, Sn = Sum of n terms,
1. Tn = a + (n – 1)d
2. Sn =
In the formula above, would give the average of the terms of the
progression. So the sum of the series is average of all the terms
multiplied by number of terms.
Substituting, l = a + (n – 1)d
We also have Sn = n/2 [2a + (n – 1)d]
If to each term of an A.P. a fixed non-zero number is added, then
the resulting progression is also an A.P.
If each term of a given A.P. is multiplied or divided by a given
non-zero fixed number k, then the resulting progression is an
A.P.
Arithmetic mean
If a and b are any two numbers, n is called the arithmetic mean
of a and b and is given by n = .
Arithmetic mean can also be found if there are more than two terms.
For instance, the arithmetic mean of a, b, c and d is equal to
(a + b + c + d)/4.
In problems, Three numbers in an AP should be taken as a – d,
a, a + d.
Four numbers in an AP should be taken as a – 3d, a – d, a + d,
a + 3d.
This would make sum of three numbers in one variable so that we
can solve it easily.
Other topics covered
Number Theory
Basic
Arithmetic Operations
Properties
of Numbers
Variable
Inconsistance
Ratio
Proportion
Percentage
Profit
and Loss
Calendar
Average
Simple
and Compound Interest
Time,
Speed and Distance
Work
and Time
Indices and Surds
Progression
Quadratic
Functions
Logarithms
Binomial Theorem
Plain Geometry
Circular Geometry
Solid Geometry
Permutations and Combinations
Probability
Set Theory
Number Systems
Practice Paper
Easy Moderate
SET Click
here SET
Click
here
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