Tips and Tricks: Arithmetic Progressions | Quantitative Aptitude for SSC CGL PDF Download

Arithmetic Progression Tricks

For an arithmetic progression with an initial term 'a' and a common difference 'd' between successive members, the n-th term of the sequence is expressed as
a_n = a+(n−1)d, where ,…n = 1,2,… and so forth.

Tips and Tricks for Arithmetic Progression To Solve Arithmetic Progression in Aptitude 

An Arithmetic Progression, also known as an Arithmetic Sequence, is a sequence of numbers or terms where the difference between consecutive terms remains constant. Below are some straightforward tips and tricks to help you swiftly, easily, and efficiently solve Arithmetic Progression questions in competitive exams.

An Arithmetic Progression is Represented in the form a, (a + d), (a + 2d), (a + 3d), …
where a = the first term, and d = the common difference.
n is number of Terms
General form, T_n = a + (n-1)d
Where T_n is nth term of an Arithmetic Progression

Properties of Arithmetic Progression

• If a fixed number is added or subtracted from each term of an AP, then the resulting sequence is also an AP and it has the same common difference as that of the original AP.
• If each term in an AP is divided or multiply with a constant non-zero number, then the resulting sequence is also in an AP.
• If n^th term is in linear expression then the sequence is in AP.
• If a₁, a₂, a₃, …, a_n and b₁, b₂, b₃, …, b_n, are in AP. then a₁+b₁, a₂+b₂, a₃+b₃, ……, a_n+b_n  and a₁–b₁, a₂–b₂, a₃–b₃, ……, a_n–b_n will also be in AP.
• If n^th term of a series is T_n = An + B, then the series is in AP
• Three terms of the A.P whose sum or product is given should be assumed as a-d, a, a+d.
• Four terms of the A.P. whose sum or product is given should be assumed as a-3d, a-d, a+d, a+3d.

Type 1: Find nth term of series tn = a + (n − 1)d

t_n = a + (n – 1)d
where t_n = nth term,
a= the first term ,
d= common difference,
n = number of terms in the sequence
In the given series,
a (first term) = 1
d (common difference) = 2 (3 – 1, 5 – 3)
Therefore, 10^th term = t₁₀ = a + (n-1) d

t₁₀ = 1 + (10 – 1) 2
t₁₀ = 1 + 18
t₁₀ = 19

Example 2: Find last term in the series if there are 8 term in this series 13 , 17 , 21 ,25….
(a) 33
(b) 41
(c) 37
(d) 39
Ans: (b)
We know that,
t_n = a + (n – 1)d
where t_n = nth term,
a = the first term ,
d = common difference,
n = number of terms in the sequence
In the given series,
a (first term) = 13
d (common difference) = 4(17 – 13, 21 – 17)
Therefore, 8^th term = t₈ = a + (n-1) d
t₈ = 13 + (8 – 1) 4
t₈ = 13 + 28
t₈ = 41

Type 2: Find number of terms in the series n =

Example 1: Find the number of terms in the series 7, 11, 15, . . .71
(a) 12
(b) 25
(c) 22
(d) 17
Ans: (d)
We know that,
where n = number of terms,
a= the first term,
l = last term,
d= common difference
In the given series,
a (first term) = 7
l (last term) = 71
d (common difference) = 11 – 7 = 4


n = 16 + 1
n = 17

Example 2: Find the number of terms if First term = 22 ,Last term = 50 and common difference is 4
(a) 10
(b) 9
(c) 8
(d) 7
Ans: (c)
We know that,

where n = number of terms,
a = the first term,
l = last term,
d = common difference
In the given series,
a (first term) = 22
l (last term) = 50
d (common difference) = 4


n = 7+ 1
n = 8

Type 3: Find sum of first ‘n’ terms of the series

Example 1: Find the sum of the series 1, 3, 5, 7…. 201
(a) 12101
(b) 25201
(c) 22101
(d) 10201
Ans: (d)
We know that,

OR

where,
a = the first term,
d= common difference,
l = tn = nth term = a + (n-1)d
In the given series,
a = 1, d = 2, and l = 201
Since we know that, l = a + (n – 1) d
201 = 1 + (n – 1) 2
201 = 1 + 2n -2
202 = 2n
n = 101


S_n = 50.5 (1 + 201)
S_n = 50.5 x 202
S_n = 10201

Example 2: Find the sum of the Arithmetic series if First term of this series is 45 , common difference is 5 and number of terms in this series is 8.
(a) 500
(b) 300
(c) 400
(d) 200
Ans: (a)
We know that,

OR

where, a = the first term,
d = common difference,
l = tn = nth term = a + (n-1)d
In the given series,
a = 45, d = 5, and n = 8


S_n = 4 (90 + 35)
S_n = 4 x 125
S_n = 500

Type 4: Find the arithmetic mean of the series. b =

Example 1: Find the arithmetic mean of first five prime numbers.
(a) 6.6
(b) 3.6
(c) 5.6
(d) 7.6
Ans: (c)
We know that

Here, five prime numbers are 2, 3, 5, 7 and 11
Therefore, their arithmetic mean (AM)

Example 2: Find Second Number if arithmetic mean of two numbers is 24 and First number is 10.
(a) 32
(b) 38
(c) 34
(d) 36
Ans: (b)
We know that

Let second Number is b
Therefore, their arithmetic mean (AM)
b = 38