Important Formulas: Arithmetic Progressions | Quantitative Aptitude for SSC CGL PDF Download

Arithmetic Progression (AP)

An arithmetic progression, also known as an arithmetic sequence, is a sequence of numbers in which the difference between any two consecutive members remains constant.

Basic Concept on Arithmetic Progression

First term is denoted by a Common difference is denoted by d n^th term is denoted by
a_n or t_n 
Sum of First n terms is denoted by S_n
Example : 4,8,12,16……..

Formula of Arithmetic Progression

nth term of an AP
Formula to find the nth term of an AP is
T_n = a + (n – 1) d
where t_n = nth term,
a = first term ,
d = common difference,
n = number of terms in the sequence.

Number of terms in an AP

Formula to find the numbers of term of an AP is

where
n = number of terms,
a = first term,
l = last term,
d= common difference.

Sum of first n terms in an AP

Formula to find the sum of first n terms of an AP is

or

where,
a = first term,
d = common difference,
t_n = n^th term = a + (n-1)d

Arithmetic Mean

If a, b, c are in AP, then the Arithmetic mean of a and c  is b  i.e.

Some other important formulas of Arithmetic Progression

• Sum of first n natural numbers
  We derive the formula to find the sum of first n natural numbers
  
  where
  S = Sum of first n natural numbers
  n = number of First n natural numbers

Sum of squares of first n natural numbers

Formula to find the sum of squares of first n natural numbers is

where
S = Sum of Squares of first n natural numbers
n = number of First n natural numbers.

Sum of first n odd numbers

• Formula to Find the Sum of First n odd numbers
  S = n²
  where
  S = Sum of first n odd  numbers
  n = number of First n odd numbers.

Sum of first n even numbers

• Formula to find the Sum of First n Even numbers is
  S = n(n+1)
  where
  S = Sum of first n Even numbers
  n = number of First n Even numbers.

Using Formulas of Arithmetic Progression in Questions

Q1: The sum of the first 20 terms of an arithmetic progression is 610, and the sum of the next 30 terms is 2130. What is the common difference of this arithmetic progression?
(a) 8
(b) 10
(c) 12
(d) 15
Ans: (b)
Let the first term of the arithmetic progression be ‘a’ and the common difference be ‘d’. The sum of the first 20 terms is given by:

The sum of the next 30 terms is:

Given S₁ = 610 and S₂ = 2130, we can write two equations:
20a + 190d = 610
30a + 590d = 2130
Solving these equations, we get d = 10. Therefore, the correct answer is (b) 10.

Q2: The sum of the first 50 terms of an arithmetic progression is 4800. If the first term is 12 and the common difference is -3, find the 30th term.
(a) -14
(b) -17
(c) -20
(d) -75
Ans: (d)
The sum of the first 50 terms of the arithmetic progression is given by:

Solving, we get S₅₀ = 4800.
Now, to find the 30th term, we can use the formula for the nth term of an AP:
a₃₀ = a + (n-1) * d
a₃₀ = 12 + (30-1) * (-3)
a₃₀ = 12 – 87
a₃₀ = -75.
The correct answer is -75. 

Q3: The 10th term of an arithmetic progression is equal to three times the 6th term. If the sum of the first 10 terms is 220, what is the common difference of the progression?
(a) 6
(b) 8
(c) 10
(d) 12
Ans: (b)
Let the first term of the arithmetic progression be ‘a’, and the common difference be ‘d’.
According to the problem, a + 9d = 3(a + 5d)
Solving this equation, we get a = 10d.
Now, the sum of the first 10 terms of the arithmetic progression is given by:

Solving this equation, we get d = 8.
Therefore, the correct answer is (B) 8.

Q4: In an arithmetic progression, the 15th term is 8 more than the 8th term. If the common difference is 3, what is the 10th term of the progression?
(a) 21
(b) 23
(c) 25
(d) 40
Ans: (d)
Let the first term of the arithmetic progression be ‘a’. Then, the 8th term is given by a + 7 * 3 = a + 21, and the 15th term is a + 14 * 3 = a + 42.
According to the problem, a + 42 = (a + 21) + 8
Solving this equation, we get a = 13.
Now, the 10th term is a + 9 * 3 = 13 + 27 = 40.
The correct answer is 40.

Q5: If the sum of the first ‘n’ terms of an arithmetic progression is given by S_n = 5n₂ + 3_n, what is the first term of the progression?
(a) 2
(b) 5
(c) 7
(d) 4
Ans: (d)
The sum of the first ‘n’ terms of an arithmetic progression is given by:

Comparing it with Sn = 5n² + 3n, we get 2a + (n-1) * d = 5n + 3
Since we are looking for the first term ‘a’, we can consider ‘n’ as 1:
2a + (1-1) * d = 5 + 3
2a = 8
a = 4.
The correct answer is 4.