In a geometric progression the 3rd term is 4 times the 1st term and the sum of the 2nd term and the 4th term is 30. Find the positive common ratio, the first term, and sum of the first 7 terms
Why did the geometric progression go to samba lessons?
Because they needed to find their common rhythm!
Now, let's solve the problem. Let the first term be "a" and the common ratio be "r".
According to the problem, the 3rd term is 4 times the 1st term, so we can write:
ar^2 = 4a
Canceling the "a" on both sides, we get:
r^2 = 4
Now, let's look at the sum of the 2nd term and the 4th term, which is given as 30:
ar + ar^3 = 30
Since we know r^2 = 4, we can substitute this into the equation:
ar + a(4r) = 30
Simplifying the equation, we get:
ar + 4ar = 30
5ar = 30
Dividing both sides by 5ar, we get:
r = 6/a
Now, substituting this value of r into r^2 = 4, we have:
(6/a)^2 = 4
36/a^2 = 4
Multiplying both sides by a^2, we get:
36 = 4a^2
Dividing both sides by 4, we have:
9 = a^2
Taking the square root of both sides, we get:
a = ±3
Since we are looking for the positive common ratio, let's take a = 3.
So, the positive common ratio is 6/3 = 2.
Now, let's find the sum of the first 7 terms. The formula for the sum of the first n terms of a geometric progression is given by:
S = a(1 - r^n) / (1 - r)
Substituting the values of a and r, we have:
S = 3(1 - 2^7) / (1 - 2)
Simplifying this equation, we get:
S = 3(1 - 128) / (-1)
S = 3(-127) / (-1)
S = 381
So, the sum of the first 7 terms is 381.
I hope my answer didn't leave you feeling too acute!
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3
To find the positive common ratio, the first term, and the sum of the first 7 terms in a geometric progression, we can use the given information about the 3rd term and the sum of the 2nd and 4th terms.
Let's begin by assigning variables to the unknowns. Let's say the first term is "a" and the common ratio is "r".
According to the given information, the 3rd term is 4 times the 1st term. This can be expressed as:
a * r^2 = 4a ----(1)
Similarly, the sum of the 2nd term and the 4th term is 30:
ar + ar^3 = 30 ----(2)
We now have two equations with two variables. We can solve for "r" and "a" simultaneously.
Let's start by dividing equation (2) by the equation (1):
(ar + ar^3) / (a * r^2) = 30 / 4
Simplifying the equation:
r + r^3 = 7.5 ----(3)
Now, let's substitute r = x in equation (3) for easier calculation, where x is the cubic root of r:
x + x^3 = 7.5
Rearranging the equation:
x^3 + x - 7.5 = 0
We need to find the positive real value of "x" that satisfies this equation. We can use numerical methods or calculators to find the value of "x" which is approximately 1.705.
Now that we have the value of "x", we can find the positive common ratio "r" by raising "x" to the power of 3:
r = x^3 = (1.705)^3 = 5.937
Next, we can substitute the value of "r" back into equation (1) to find the value of "a":
a * (5.937)^2 = 4a
Simplifying the equation:
35.218a = 4a
Dividing both sides by "a":
35.218 = 4
Dividing both sides by 4:
a = 0.113
So, the positive common ratio (r) is approximately 5.937, the first term (a) is approximately 0.113.
To find the sum of the first 7 terms in the geometric progression, we can use the formula:
Sum of n terms = a * (r^n - 1) / (r - 1)
Substituting the values we found, we have:
Sum of 7 terms = 0.113 * (5.937^7 - 1) / (5.937 - 1)
Evaluating the equation, we find that the sum of the first 7 terms is approximately 38.968.
r^(3 - 1) = 4 ... r = 2
2nd term ... t + 4 t = 30 ... t = 6
1st term ... 6 / 2 = ?
write the 1st seven terms and sum them