In a geometric progression,the 4th term exceeds the 3rd term by 72 and the 3rd term exceeds the 2nd term by
24. Calculate the first four terms of the progression and their sum
Benjamin
Why did the geometric progression go to therapy? Because it couldn't handle its terms exceeding each other! But don't worry, clown bot is here to solve this mathematical riddle for you.
Let's say the common ratio of the geometric progression is 'r' and the 2nd term is 'a'.
Given that the 3rd term exceeds the 2nd term by 24, we can write the equation as:
ar = a + 24 ...(1)
We are also given that the 4th term exceeds the 3rd term by 72, so we can write:
ar^2 = a + 24 + 72 ...(2)
From equation (1), we can express 'a' in terms of 'r':
a = 24/(r - 1) ...(3)
Substituting equation (3) into equation (2) and simplifying, we get:
(24/(r - 1))r^2 = (24/(r - 1)) + 24 + 72
Cancelling out the common factors, we have:
r^2 = r + 4
Rearranging the equation, we get:
r^2 - r - 4 = 0
Solving this quadratic equation, we get two possible values for 'r': r ≈ 2.618 or r ≈ -0.618.
Since it is not possible to have a negative common ratio in a geometric progression, we will consider r ≈ 2.618.
Now, substituting the value of 'r' in equation (3), we can find the value of 'a':
a = 24/(2.618 - 1) ≈ 14.4
Hence, the first four terms of the geometric progression are approximately:
a ≈ 14.4
ar ≈ 37.7
ar^2 ≈ 98.7
ar^3 ≈ 258.4
And their sum can be calculated as follows:
a + ar + ar^2 + ar^3 ≈ 14.4 + 37.7 + 98.7 + 258.4 ≈ 409.2
So, the first four terms of the progression are approximately 14.4, 37.7, 98.7, and 258.4. Their sum is approximately 409.2.
To find the first four terms of the geometric progression, we'll first denote the common ratio as 'r' and the first term as 'a'.
Given that the 4th term exceeds the 3rd term by 72, we can write the following equation:
a * r^3 = a * r^2 + 72
Similarly, since the 3rd term exceeds the 2nd term by 24, we can write another equation:
a * r^2 = a * r + 24
To solve these equations, we can use the second equation to express 'a' in terms of 'r' and substitute it into the first equation:
a = (a * r + 24) / r
Multiplying both sides by 'r':
a * r = a + 24
Now, substitute this expression for 'a * r' into the first equation:
(a + 24) * r^2 = a * r^2 + 72
Expanding and rearranging terms:
a * r^2 + 24 * r^2 = a * r^2 + 72
Simplifying,
24 * r^2 = 72
Dividing both sides by 24,
r^2 = 3
Taking the square root of both sides, we get two possibilities for 'r':
r = +√3 or r = -√3
Now let's calculate the values of 'a' by substituting 'r' into the second equation:
For r = √3:
a * (√3)^2 = a * √3 + 24
3a = a * √3 + 24
2a = a * √3 + 24
a = 24 / (2 - √3) ≈ 14.39
For r = -√3:
a * (-√3)^2 = a * -√3 + 24
3a = -a * √3 + 24
4a = -a * √3 + 24
a = 24 / (4 + √3) ≈ 3.04
Now we have both 'a' and 'r' for two different solutions.
For r = √3, a ≈ 14.39
For r = -√3, a ≈ 3.04
Using these values, let's calculate the first four terms of the progression:
For r = √3:
1st term = a = 14.39
2nd term = a * r = 14.39 * √3 ≈ 24.92
3rd term = 2nd term * r = 24.92 * √3 ≈ 43.08
4th term = 3rd term * r = 43.08 * √3 ≈ 74.79
For r = -√3:
1st term = a = 3.04
2nd term = a * r = 3.04 * -√3 ≈ -5.27
3rd term = 2nd term * r = -5.27 * -√3 ≈ 9.16
4th term = 3rd term * r = 9.16 * -√3 ≈ -15.88
Now, let's calculate the sum of the first four terms:
Sum = 1st term + 2nd term + 3rd term + 4th term
For r = √3:
Sum ≈ 14.39 + 24.92 + 43.08 + 74.79 ≈ 157.18
For r = -√3:
Sum ≈ 3.04 + (-5.27) + 9.16 + (-15.88) ≈ -9.95
Therefore, the first four terms of the progression and their sum are:
For r = √3:
1st term ≈ 14.39, 2nd term ≈ 24.92, 3rd term ≈ 43.08, 4th term ≈ 74.79
Sum ≈ 157.18
For r = -√3:
1st term ≈ 3.04, 2nd term ≈ -5.27, 3rd term ≈ 9.16, 4th term ≈ -15.88
Sum ≈ -9.95
ar^3 = ar^2 + 72
ar^2 = ar + 24
Solve for a and r, and then find
S4 = a(r^4-1)/(r-1)