The sum of the first two terms of G.P is 30. The sum of infinity terms of G.P is 40. What is the common ratio?
Answers
Topic :-
- Geometric Progression
Given :-
- The sum of the first two terms of a geometric progression is 30.
- The sum of the infinity terms of a geometric progression is 40.
To Find :-
- The common ratio of given geometric progression.
Solution :-
A progression in which ratio of its consecutive terms remains constant is known as Geometric Progression (GP).
General terms of GP :-
a, ar, ar², ar³, ar⁴, . . . . . . . , arⁿ⁻¹
where
a is first term
r is common ratio and
n is order of term
It is given that,
The sum of the first two terms of a geometric progression is 30.
➟ a + ar = 30
➟ a(1 + r) = 30 . . . . . Equation (1)
It is also given that,
The sum of the infinity terms of a geometric progression is 40.
Sum of infinite terms of a GP = a/(1 - r); |r| < 1
➟ 40 = a/(1 - r)
Cross multiplying,
➟ a = 40(1 - r) . . . . . Equation (2)
Substitute value of 'a' from Equation (2) to Equation (1),
➟ a(1 + r) = 30
➟ 40(1 - r)(1 + r) = 30
➟ 40(1 - r²) = 30
(∵ (a - b)(a + b) = a² - b²)
➟ 1 - r² = 30/40
➟ r² = 1 - (3/4)
➟ r² = (4 - 3)/4
➟ r² = 1/4
➟ r = ± 1/2
Answer :-
- So, the common ratio of the GP can be 1/2 and -1/2 both.