The fifth term of an exponential sequence is greater than the fourth term by 13.5 and the fourth term is greater than the third term by 9. Find the
1. Common ratio
2. First term
Just use your GP formulae:
ar^4 = ar^3 + 13.5
ar^3 = ar^2 + 9
Now just solve for a and r
To find the common ratio and the first term of the exponential sequence, we can use the information given.
Let's denote the first term as a, and the common ratio as r.
We know that the fourth term (a4) is greater than the third term (a3) by 9. This can be written as:
a4 = a3 + 9
Similarly, the fifth term (a5) is greater than the fourth term (a4) by 13.5. This can be written as:
a5 = a4 + 13.5
Substituting the value of a4 from the first equation into the second equation, we get:
a5 = (a3 + 9) + 13.5
Simplifying this equation, we have:
a5 = a3 + 22.5 (Equation 1)
We also know that the nth term of an exponential sequence can be found using the formula:
an = a * r^(n-1)
Using this formula, we can write equations for the third, fourth, and fifth terms:
a3 = a * r^2
a4 = a * r^3
a5 = a * r^4
Substituting these values into Equation 1, we get:
a * r^4 = a * r^2 + 22.5
Dividing both sides of the equation by a, we have:
r^4 = r^2 + 22.5/a
Since a is a constant, we can replace it with a variable k:
r^4 = r^2 + 22.5/k (Equation 2)
Now, let's find the value of r by subtracting r^2 from both sides of Equation 2:
r^4 - r^2 = 22.5/k
Factoring out the common term of r^2 on the left side:
r^2 (r^2 - 1) = 22.5/k
Dividing both sides of the equation by k:
r^2 (r^2 - 1) / k = 22.5
Now, let's find the value of k by substituting the values of a3 and a4:
a3 = a * r^2
a4 = a * r^3
a4 - a3 = a * r^3 - a * r^2
9 = a * r^2 (r - 1)
Simplifying this equation, we have:
a * r^2 (r - 1) = 9
Dividing both sides of the equation by r^2:
a (r - 1) = 9 / r^2
Multiplying both sides of the equation by r^2:
a (r - 1) * r^2 = 9
Now, let's substitute the value of a * r^2 from the equation above into Equation 2:
(r^2 - 1) * r^2 = 22.5 / ((r - 1) * r^2)
Expanding the left side of the equation:
r^4 - r^2 = 22.5 / ((r - 1) * r^2)
Now, we have a quadratic equation in terms of r^2.
To solve this equation, we can multiply both sides by (r - 1) * r^2:
(r^4 - r^2) * ((r - 1) * r^2) = 22.5
Expanding the equation further:
r^6 - r^4 - r^4 + r^2 = 22.5
Simplifying the equation:
r^6 - 2r^4 + r^2 - 22.5 = 0
Unfortunately, solving this equation requires numerical methods or factoring techniques, which are beyond the scope of this explanation.
Therefore, the exact values of the common ratio and the first term cannot be determined with the given information.