Sequence & Series For SSC CGL Questions With Solutions

SSC CGL Sequence & Series Important Formulas

S.No.

Formula / Concept

Expression

1

Sum of the first n natural numbers

1+2+3+⋯+n=n(n+1)21 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}

2

Sum of the first n even natural numbers

2+4+6+⋯+2n=n(n+1)2 + 4 + 6 + \dots + 2n = n(n+1)

3

Sum of the first n odd natural numbers

1+3+5+⋯+(2n−1)=n21 + 3 + 5 + \dots + (2n-1) = n^2

4

Sum of squares of the first n natural numbers

12+22+32+⋯+n2=n(n+1)(2n+1)61^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}

5

Sum of squares of the first n even natural numbers

22+42+62+⋯+(2n)2=2n(n+1)(2n+1)32^2 + 4^2 + 6^2 + \dots + (2n)^2 = \frac{2n(n+1)(2n+1)}{3}

6

Sum of squares of the first n odd natural numbers

12+32+52+⋯+(2n−1)2=n(2n−1)(2n+1)31^2 + 3^2 + 5^2 + \dots + (2n-1)^2 = \frac{n(2n-1)(2n+1)}{3}

7

Sum of cubes of the first n natural numbers

13+23+33+⋯+n3=(n(n+1)2)21^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n+1)}{2}\right)^2

8

Sum of cubes of the first n even natural numbers

23+43+63+⋯+(2n)3=2[n(n+1)]22^3 + 4^3 + 6^3 + \dots + (2n)^3 = 2[n(n+1)]^2

9

Sum of cubes of the first n odd natural numbers

13+33+53+⋯+(2n−1)3=n2(2n2−1)1^3 + 3^3 + 5^3 + \dots + (2n-1)^3 = n^2(2n^2 - 1)

10

nth term of an Arithmetic Progression (AP)

Tn=a+(n−1)dT_n = a + (n-1)d

11

Sum of the first n terms of an AP

Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d] or Sn=n2(a+l)S_n = \frac{n}{2}(a + l)

12

nth term of a Geometric Progression (GP)

Tn=arn−1T_n = ar^{n-1}

13

Sum of the first n terms of a GP

Sn=a(rn−1)r−1,  r≠1S_n = \frac{a(r^n - 1)}{r-1}, \; r \neq 1

14

Sum of infinite terms of a GP (if

r

15

Arithmetic Mean (AM) of two numbers

AM=a+b2\text{AM} = \frac{a+b}{2}

16

Arithmetic Mean (AM) of n numbers

AM=a1+a2+⋯+ann\text{AM} = \frac{a_1 + a_2 + \dots + a_n}{n}

17

Geometric Mean (GM) of two numbers

GM=ab\text{GM} = \sqrt{ab}

18

Geometric Mean (GM) of n numbers

GM=(a1a2a3…an)1n\text{GM} = (a_1a_2a_3\dots a_n)^{\tfrac{1}{n}}

19

Relation between AM and GM (for positive numbers)

AM≥GMAM \geq GM

20

Relation between AM and GM (for negative numbers)

AM≤GMAM \leq GM