Some Shortcuts in Mathematics
PROGRESSION
Arithmetic Progression (A.P.):
It is a series in which any two consecutive terms have a common difference and the next term can be derived
by adding that common difference to the previous term.
Therefore Tn+1 – Tn = constant and called common difference (d) for all n N.
If in an A. P. a = first term,
d = common difference = Tn – Tn-1
Tn = nth term (Thus T1 = first term, T2 = second term, T10 tenth term and so on.)
l = last term,
Sn = Sum of n terms.
a, a + d, a + 2d, a + 3d,... are in A.P.
The n t h term of an A.P is given by the formula
Note: If the last term of the A.P. consisting of n terms be l , l = a + (n – 1) d
The sum of first n terms of an AP is usually denoted by Sn and is given by the following formula:
Where ‘ l ’ is the last term of the series.
Properties of an AP:
I. If each term of an AP is increased, decreased, multiplied or divided by the same non-zero number, the resulting sequence is also an AP.
For example: For A.P. 3, 5, 7, 9, 11…
If you add a constant let us say 1 to each term, you get 4, 6, 8, 10, 12......
This is an A.P. with a common difference of 2.
If you multiply each term by a constant let us say 2, you get 6, 10, 14, 18, 22…..
Again this is an A.P. with a common difference of 4.
n = a + (n – 1) d
Note: If the last term of the A.P. consisting of n terms be l ,
last term = l = a + (n – 1) d
The sum of first n terms of an AP is usually denoted by Sn and is given by the following formula:
Arithmetic Progression (A.P.):
It is a series in which any two consecutive terms have a common difference and the next term can be derived
by adding that common difference to the previous term.
Therefore Tn+1 – Tn = constant and called common difference (d) for all n N.
If in an A. P. a = first term,
d = common difference = Tn – Tn-1
Tn = nth term (Thus T1 = first term, T2 = second term, T10 tenth term and so on.)
l = last term,
Sn = Sum of n terms.
a, a + d, a + 2d, a + 3d,... are in A.P.
The n t h term of an A.P is given by the formula
Note: If the last term of the A.P. consisting of n terms be l , l = a + (n – 1) d
The sum of first n terms of an AP is usually denoted by Sn and is given by the following formula:
Where ‘ l ’ is the last term of the series.
Properties of an AP:
I. If each term of an AP is increased, decreased, multiplied or divided by the same non-zero number, the resulting sequence is also an AP.
For example: For A.P. 3, 5, 7, 9, 11…
If you add a constant let us say 1 to each term, you get 4, 6, 8, 10, 12......
This is an A.P. with a common difference of 2.
If you multiply each term by a constant let us say 2, you get 6, 10, 14, 18, 22…..
Again this is an A.P. with a common difference of 4.
n = a + (n – 1) d
Note: If the last term of the A.P. consisting of n terms be l ,
last term = l = a + (n – 1) d
The sum of first n terms of an AP is usually denoted by Sn and is given by the following formula:
Properties of an AP:
I. If each term of an AP is increased, decreased, multiplied or divided by the same non-zero number, the resulting sequence is also an AP.
For example: For A.P. 3, 5, 7, 9, 11…
II. In an AP, the sum of terms equidistant from the beginning and end is always same and equal to the sum
of first and last terms as shown in example below.
III. Three numbers in AP are taken as a – d, a, a + d.
For 4 numbers in AP are taken as a – 3d, a – d, a + d, a + 3d
For 5 numbers in AP are taken as a – 2d, a – d, a, a + d, a + 2d
IV. Three numbers a, b, c are in A.P. if 2b = a + c. or b is called the Arithmetic mean of a & c
Geometric Progression:
A series in which each preceding term is formed by multiplying it by a constant factor is called a Geometric
Progression G. P. The constant factor is called the common ratio and is formed by dividing any term by the
term which precedes it.
In other words, a sequence, a1, a2, a3, …, an,… is called a geometric progression
If ( a n+1 )/ ( a n ) = constant for all natural numbers
The General form of a G. P. with n terms is a, ar, ar^2…ar^n –1
Thus, if a = the first term
r = the common ratio,
Tn = nth term and
Sn = sum of n terms
I. If each term of an AP is increased, decreased, multiplied or divided by the same non-zero number, the resulting sequence is also an AP.
For example: For A.P. 3, 5, 7, 9, 11…
II. In an AP, the sum of terms equidistant from the beginning and end is always same and equal to the sum
of first and last terms as shown in example below.
III. Three numbers in AP are taken as a – d, a, a + d.
For 4 numbers in AP are taken as a – 3d, a – d, a + d, a + 3d
For 5 numbers in AP are taken as a – 2d, a – d, a, a + d, a + 2d
IV. Three numbers a, b, c are in A.P. if 2b = a + c. or b is called the Arithmetic mean of a & c
Geometric Progression:
A series in which each preceding term is formed by multiplying it by a constant factor is called a Geometric
Progression G. P. The constant factor is called the common ratio and is formed by dividing any term by the
term which precedes it.
In other words, a sequence, a1, a2, a3, …, an,… is called a geometric progression
If ( a n+1 )/ ( a n ) = constant for all natural numbers
The General form of a G. P. with n terms is a, ar, ar^2…ar^n –1
Thus, if a = the first term
r = the common ratio,
Tn = nth term and
Sn = sum of n terms