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

Geometric Progression

A Geometric Progression is a sequence of numbers in which each term, except the first, is determined by multiplying the preceding one by a constant, non-zero number known as the common ratio.

Formulas for GP

• Common ratio:
  Formula for finding the common ratio a2/a1
• nth term of an GP:
  Formula for finding the nth term of an GP a_n = a₁r^n-1  where a₁ =  First Term,
  r = common ratio and
  n = number of Terms
• Sum of first n terms in an GP:
  Standard Formula for sum of first n terms in an GP
    {if r>1} where, r = common ratio, a₁ = First Term, n = number of terms
    {if r<1} where, r = common ratio, a₁ = First Term, n = number of terms
• Sum of an infinite GP :
  Formula for Sum of an infinite GP
    (if  -1<r<1)
• Geometric Mean (GM): 
  If two non-zero numbers a and b are in GP, then there GM is GM = (ab)^1/2
  If three non-zero numbers a,b and c are in GP, then there GM is GM = (abc)^1/3    

Properties of Geometric Progression

• If ‘a’ is the first term, r is the common ratio of a finite G.P. consisting of m terms, then the nth term from the end will be =  a r^m-n.
• The nth term from the end of the G.P. with the last term ‘l’ and common ratio r is
• Reciprocal of all the term in G.P are also considered in the form of G.P.
• When all terms is GP raised to same power, the new series of geometric progression is form.

Using Formulas of Geometric Progression in Questions

Q1: What is the 7th term of a geometric progression if the first term (a) is 3 and the common ratio (r) is -2?
(a) -192
(b) 192
(c) -96
(d) 96
Ans: (b)
The nth term of a GP is given by an = a * r^(n-1). Substituting the values, a = 3, r = -2, and n = 7:
a7 = 3 * (-2)^(7-1) = 3 * (-2)⁶ = 3 * 64 = 192

Q2: If the sum of an infinite geometric progression is 72 and the common ratio (r) is 0.5, what is the first term (a)?
(a) 2
(b) 12
(c) 24
(d) 36
Ans: (d)
The sum of an infinite GP is given by S = a / (1 – r). Substituting the values, S = 72 and r = 0.5:
72 = a / (1 – 0.5)
72 = a / 0.5
a = 72 * 0.5
a = 36

Q3: Find the common ratio (r) of a geometric progression if the sum of the first 8 terms (S8) is 4374 and the first term (a) is 3.
(a) 3
(b) 2
(c) 4
(d) 5
Ans: (b)
The sum of the first n terms of a GP is given by Sn = a * (rⁿ – 1) / (r – 1). Substituting the values, a = 3, S8 = 4374, and n = 8:
4374 = 3 * (r⁸ – 1) / (r – 1)
Now, solving for r:
3 * (r⁸ – 1) = 4374 * (r – 1)
r⁸ – 1 = 1458 * (r – 1)
r⁸ – 1458r + 1457 = 0
Using numerical methods, we find that r ≈ 2.362

Q4: What is the sum of the first 5 terms of a geometric progression if the first term (a) is 2 and the common ratio (r) is 3/4?
(a) 6.75
(b) 8.25
(c) 7.50
(d) 5.50
Ans: (c)
The sum of the first n terms of a GP is given by Sn = a * (r^n – 1) / (r – 1). Substituting the values, a = 2, r = 3/4, and n = 5:
S5 = 2 * ((3/4)⁵ – 1) / (3/4 – 1)
= 2 * (243/1024 – 1) / (-1/4)
= 2 * (-781/1024) / (-1/4) = 7.50

Q5: What is the 10th term of a geometric progression if the 4th term is 54 and the common ratio (r) is 3?
(a) 1458
(b) 2187
(c) 729
(d) None of the above
Ans: (d)
The nth term of a GP is given by an = a * r^(n-1).
Substituting the values, a = 54, r = 3, and n = 10:
a10 = 54 * 3^(10-1) = 54 * 3⁹ = 54 * 19683 = 1058841