The first term of a geometric progression is more than the third term by 12. The fourth term is more than the second term by 4. Find:
i.the first term a and the common ratio r.
ii. the n'th term of the progression.
Geometric Progression: x(n) = a r^n
First term x(0) = a
Second term x(1) = ar
etc.
So
a = ar^2 +12
ar^3 = ar +4
Rearranging gives,
a(1-r^2) = 12
a(1-r^2)r = -4
Thus solve for r by dividing,
Substitute into the original to solve for a,
Omg im really sorry. There seems to be an error. The "fifth" term of a geometric progression is more than the third term by 12.
To solve this problem, we need to use the information given and apply the formulas for geometric progressions. Let's start by writing down what we know.
Given:
1. The first term is more than the third term by 12, which can be expressed as: a - ar^2 = a + 12, where a is the first term and r is the common ratio.
2. The fourth term is more than the second term by 4, which can be expressed as: ar^3 - ar = ar^2 + 4.
Now let's solve for a and r using these equations:
1. a - ar^2 = a + 12
Subtracting a from both sides, we get:
-ar^2 = 12
Dividing both sides by -1, we find:
ar^2 = -12
2. ar^3 - ar = ar^2 + 4
Simplifying, we get:
ar^3 - ar^2 - ar - 4 = 0
Now we can use these two equations to find the values of a and r.
To solve for a:
From the equation ar^2 = -12, we can substitute this into the second equation:
-a - ar - ar^3 + 4 = 0
Simplifying further:
-a(1 + r + r^3) + 4 = 0
-a(r^3 + r + 1) + 4 = 0
Since we know that r^3 + r + 1 cannot be zero, we can divide both sides by (r^3 + r + 1):
-a = -4 / (r^3 + r + 1)
a = 4 / (r^3 + r + 1)
Now let's solve for r:
From the first equation, we have ar^2 = -12. We can substitute a using the value we found above:
(4 / (r^3 + r + 1)) * r^2 = -12
Simplifying further:
4r^2 / (r^3 + r + 1) = -12
Multiplying both sides by (r^3 + r + 1):
4r^2 = -12(r^3 + r + 1)
4r^2 = -12r^3 - 12r - 12
Rearranging to form a cubic equation:
12r^3 + 4r^2 + 12r + 12 = 0
Now, solving this cubic equation is beyond the scope of this explanation, as it requires advanced algebraic methods. To find the values of a and r, you can use numerical approximation methods such as the Newton-Raphson method or use a graphing calculator to find the intersection points of the equation with the x-axis.
Once you have found the values of a and r, you can use the formulas for geometric progressions to find the nth term of the progression.
The formula for the nth term of a geometric progression is: an = a * r^(n-1)
Substitute the values of a and r that you found into this formula to calculate the desired nth term.