Sequences And Series Of Numbers
Sequence: A succession of numbers arranged in a definite order according to a certain rule
is called a sequence. A sequence is said to be finite or infinite according as the number of
terms in it is finite or infinite respectively. The number occurring at the nth place of a
sequence is called its general term (or nth term) and is denoted by Tn .
Series: By adding the terms of a sequence, we get a series. A series is said to be finite or
infinite depending on the number of terms in it.
Progression: In mathematics, a progression is a special type of sequence for which it is
possible to obtain a formula for the nth term of the aforementioned sequence. There are
three main types of progressions. They are-
1. Arithmetic Progression (AP)
2. Geometric Progression (GP)
3. Harmonic Progression (HP)
Arithmetic Progression (AP): It is a sequence of numbers in which the difference of any two
consecutive numbers is always a constant, known as the common difference. For example,
1, 3, 5, 7, 9… is an AP with common difference 2.
Geometric Progression (GP): It is a sequence of numbers in which the ratio of any two
consecutive numbers is always a constant, known as the common ratio. In a GP each term
after the first term is found by multiplying the previous term by the common ratio. For
example, 2, 4, 8, 16, 32… is a GP with common ratio 2.
Harmonic Progression (HP): It is a sequence of numbers formed by taking the reciprocals of
1 1 1 1
an arithmetic progression. For example, 1, , , , … is a HP as 1, 2, 3, 4, 5… is an AP
2 3 4 5
with common difference 1.
For any progression, there are two main formulas we will come across. They are-
• The nth term of a progression (denoted by an )
• The sum of first n terms of the progression (denoted by S n )
Arithmetic Progression
As discussed earlier, in an AP, the difference of any two consecutive terms is always a
constant. Hence the next term of an AP is obtained by adding a fixed number, i.e.- the
common difference, to the previous term. In case of any AP, there are four main
terminologies one must be acquainted with. They are-
• First term of the AP, denoted by a.
• Common difference, denoted by d.
• nth term of the AP, denoted by an .
• Sum of first n terms of the AP, denoted by S n .
Consider the first term of an AP to be a and let the common difference be d. Consecutive
terms can be found by adding d to a in the following manner-
Position Of The Term Representation Of The Value Of The Term
Term
1 a1 a
2 a2 a+d
3 a3 a + 2d
4 a4 a + 3d
. . .
. . .
n an a + (n − 1)d
Clearly, the difference between any two terms is d which can be obtained as-
d = a2 − a1 = a3 − a2 = a4 − a3 = .... = an+1 − an , n = 1, 2, 3, …
If the value of d is positive, then the member terms will grow gradually towards positive
infinity.
If the value of d is negative, then the member terms will diminish gradually towards
negative infinity.
Formula Table:
Consider an AP with first term a and common difference d. Then-
Common Difference (d) d = an+1 − an
nth term ( an ) an = a + (n − 1)d
Sum of first n terms ( S n ) n
• S n = 2a + (n − 1)d
2
n
• S n = a + an
2
n
• Sn = a + l
2
where, l is the last term of the finite AP and n denotes the number of terms.
Geometric Progression
As discussed earlier, GP is a sequence of numbers where each term after the first is found by
multiplying the previous one by a fixed non zero number called the common ratio which is
denoted by r. In case of any GP, there are four main terminologies one must be acquainted
with. They are-
• First term of the GP, denoted by a.
• Common ratio, denoted by r.
• nth term of the GP, denoted by an .
• Sum of first n terms of the GP, denoted by S n .
Consider the first term of a GP to be a and let the common ratio be r. Consecutive terms can
be found by multiplying r to a in the following manner-
Position Of The Term Representation Of The Value Of The Term
Term
1 a1 a
2 a2 ar
3 a3 ar 2
4 a4 ar 3
. . .
. . .
n an ar n −1
Clearly, the ratio of any two terms is r which can be obtained as-
a a a a
r = 2 = 3 = 4 = ..... = n +1 0 , n = 1, 2, 3, …
a1 a2 a3 an
The behaviour of a GP depends on the value of the common ratio. If the common ratio is-
• Positive, then each term will be the same sign as the first term.
• Negative, then each term will alternate between positive and negative.
• 1, then the progression is a constant sequence.
• −1 , then the progression will have constant terms with alternating sign.
• Greater than 1, then there will be exponential growth towards positive or negative
infinity (depending on the sign of the initial term)
• Between 1 and −1 but not zero, there will be exponential decay towards zero.
Formula Table:
Consider a GP with first term a and common ratio r. Then-
Common Ratio (r) an +1
r=
an
nth term ( an ) an = ar n −1
Sum of first n terms ( S n ) a ( r n − 1)
• Sn = , if r 1
r −1
a (1 − r n )
• Sn = , if r 1
1− r
• Sn = na , if r = 1
Tips and Tricks for solving numericals:
• If three consecutive numbers are in AP, take the three numbers as a − d , a, a + d
where a is the first term and d is the common difference.
• If five consecutive numbers are in AP, take the five numbers as
a − 2d , a − d , a, a + d , a + 2d where a is the first term and d is the common difference.
a
• If three consecutive numbers are in GP, take the three numbers as , a, ar where a is
r
the first term and r is the common ratio.
a a
• If five consecutive numbers are in GP, take the five numbers as 2
, , a, ar , ar 2
r r
where a is the first term and r is the common ratio.
• The kth term from the end of an AP or GP is the (n − k + 1) th term from the beginning.
• If ak represents the kth term of an AP and S k denotes the sum of first k terms of that
AP then, am = Sm+1 − Sm for some m = 1, 2,3..., n .
