the fifth term of an exponential sequence [GP] is greater than the term by 13.5 and the fourth term is greater than the third term by 9 find the common ratio and the first term.
I will assume it says,
"the fifth term of an exponential sequence [GP] is greater than the fourth term by 13.5"
ar^4 - ar^3 = 13.5
ar^3(r - 1) = 13.5
"the fourth term is greater than the third term by 9"
ar^3 - ar^2 = 9
ar^2(r-1) = 9
divide the first equation by the second, the a's cancel, and the r-1 cancels
r = 13.5/9 = 135/90 = 3/2
back in ar^2(r-1) = 9
a(9/4)(1/2) = 9
a = 8
check: the sequence is 8, 12, 18, 27, 40.5,60.75, ....
fourth - third = 27-18 = 9
fifth - fourth = 40.5 - 27 = 13.5
all is good
change the numbers to whatever they should be, stick with the method
Let's assume that the first term of the exponential sequence is a and the common ratio is r.
We are given that the fifth term (a*r^4) is greater than the term by 13.5, which can be written as:
a*r^4 > a*r^4 - 13.5 (1)
We are also given that the fourth term (a*r^3) is greater than the third term (a*r^2) by 9, which can be written as:
a*r^3 > a*r^2 + 9 (2)
Let's solve equation (1) for 13.5:
a*r^4 - a*r^4 + 13.5 > 0
13.5 > 0
As 13.5 is greater than 0, we can conclude that the common ratio (r) must be positive.
Now, let's solve equation (2) for 9:
a*r^3 - a*r^2 - 9 > 0
Let's factor out 'a' and divide by 'r^2':
a * (r^3 - r^2) - 9 > 0
Since we know that 'a' and 'r' are positive, we can cancel out the 'a' term to get:
r^3 - r^2 - 9 > 0
We can also rewrite this equation as:
r^3 - r^2 > 9 (3)
To determine the common ratio (r) and the first term (a), we need to find the values of 'r' and 'a' that satisfy both equation (1) and equation (3).
Unfortunately, with the given information, we cannot determine the exact values of 'r' and 'a' as there are multiple possibilities that satisfy the given conditions. Further information is needed to find specific values for 'r' and 'a'.
To find the common ratio and the first term of the exponential sequence, we need to solve a system of equations using the given information.
Let's denote the first term of the sequence as 'a' and the common ratio as 'r'.
Given that the fourth term is greater than the third term by 9, we can write the equation:
ar^3 - ar^2 = 9 -----(1)
Given that the fifth term is greater than the term by 13.5, we can write the equation:
ar^4 - ar^3 = 13.5 -----(2)
Now, let's solve this system of equations.
Multiplying equation (1) by r and subtracting equation (2) from it, we get:
ar^4 - ar^3 - 9r = 0
Factoring out the common term 'r', we have:
r(ar^3 - ar^2 - 9) = 0
Since r cannot be zero, we can divide both sides of the equation by (ar^3 - ar^2 - 9) to find the value of 'r'.
ar^3 - ar^2 - 9 = 0
Next, let's substitute the value of 'r' back into either equation (1) or (2) to find the value of 'a'.
Now, to solve this equation for 'r', we can use numerical methods like trial and error, or we can use a solver to find the value of 'r'.
Once we determine the value of 'r', we can substitute it back into equation (1) or (2) to solve for 'a'.
By following these steps, you will be able to find the common ratio (r) and the first term (a) of the exponential sequence.