Table of contents  
Geometric Progression  
Formulas for GP  
Properties of Geometric Progression  
Using Formulas of Geometric Progression in Questions 
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, nonzero number known as the common ratio.
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^{(n1)}. Substituting the values, a = 3, r = 2, and n = 7:
a7 = 3 * (2)^{(71)} = 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 * (r^{n} – 1) / (r – 1). Substituting the values, a = 3, S8 = 4374, and n = 8:
4374 = 3 * (r^{8} – 1) / (r – 1)
Now, solving for r:
3 * (r^{8} – 1) = 4374 * (r – 1)
r^{8} – 1 = 1458 * (r – 1)
r^{8} – 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^{(n1)}.
Substituting the values, a = 54, r = 3, and n = 10:
a10 = 54 * 3^{(101)} = 54 * 3^{9 }= 54 * 19683 = 1058841
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