sum of the first 2 terms of an increasing GP is 20. Sum of the second term and third term of the same GP is 30. Determine the common ratio.
a(1+r) = 20
ar(1+r) = 30
now divide to get
r = 3/2
the GP is 8,12,18,...
Send the complete answer
a(1+r)=20
Divide by↓
ar(1+r)=30
answer ↓
common ratio= 1.5
So nice💯💯✌️
Can you please go by the procedure.
To find the common ratio of the geometric progression (GP), we'll use the given information.
Let's assume that the first term of the GP is "a" and the common ratio is "r".
Given information:
The sum of the first two terms of the GP is 20:
a + ar = 20
The sum of the second term and the third term of the GP is 30:
ar + ar^2 = 30
Let's solve these equations to determine the common ratio.
Simplifying the first equation, we get:
a(1 + r) = 20
a = 20 / (1 + r) -- Equation 1
Similarly, simplifying the second equation, we get:
ar(1 + r) = 30
ar = 30 / (1 + r)
ar = 30 / (r^2 + r) -- Equation 2
To find the common ratio (r), we need to eliminate "a" from the equations by substitution.
Substituting Equation 1 into Equation 2, we get:
(20 / (1 + r)) * r = 30 / (r^2 + r)
20r = 30 / (r + 1)
20r^2 + 20r = 30
20r^2 + 20r - 30 = 0
Now, we have a quadratic equation. Let's solve it using the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 20, b = 20, and c = -30.
Plugging in these values, we get:
r = (-(20) ± √((20)^2 - 4(20)(-30))) / (2(20))
r = (-20 ± √(400 + 2400)) / 40
r = (-20 ± √2800) / 40
r = (-20 ± √(4 * 700)) / 40
r = (-20 ± 2√700) / 40
r = (-5 ± √700) / 10
We have two possible values for the common ratio: (-5 + √700) / 10 and (-5 - √700) / 10.
Therefore, the common ratio of the GP can be either (-5 + √700) / 10 or (-5 - √700) / 10.