The fifth term of an exponential sequence is greater than the fourth term by 27/2 and the fourth term is greater than the third term by 9. Find the first term and the common ratio

You have several equations to work with.

We know tn=a(r)^n-1
and that
t5 = t4 + 27/2
and that
t3 = t2 - 9
we also know things like t5=ar^4
Does this inspire you to use the equations and use some of your solving methods?

Please no.further a little

Please could I get the complete solution

The fifth term of aG.P is greater than the fourth term by 13.5,and the fourth term is greater than the third by 9.find : the common ratio and common difference

Why did the exponential sequence go to therapy? Because it had an exponent-ial crisis!

Let's solve this puzzle together. We can set up some equations based on the information given.

Let's call the first term of the exponential sequence "a" and the common ratio "r". The terms of the sequence can be expressed as:

First term = a
Second term = ar
Third term = ar^2
Fourth term = ar^3
Fifth term = ar^4

We're given that the fifth term is greater than the fourth term by 27/2, so we can set up the equation:

ar^4 - ar^3 = 27/2

We're also given that the fourth term is greater than the third term by 9, so we can set up the equation:

ar^3 - ar^2 = 9

To solve for the first term (a) and the common ratio (r), we need to solve these two equations simultaneously. Let's start by simplifying them:

Equation 1: ar^4 - ar^3 = 27/2
Equation 2: ar^3 - ar^2 = 9

Let's multiply both sides of Equation 2 by r:

ar^3 - ar^2 = 9
ar^4 - ar^3 = 9r

Now we have two equations:

ar^4 - ar^3 = 27/2
ar^4 - ar^3 = 9r

Since the left sides of these equations are the same, we can set their right sides equal to each other:

27/2 = 9r

To solve for r, let's divide both sides by 9:

27/2 / 9 = r
27/18 = r
3/2 = r

We have found the common ratio (r) = 3/2.

Now let's substitute this value back into Equation 1 to find the first term (a):

(ar^4 - ar^3) = 27/2
(a(3/2)^4 - a(3/2)^3) = 27/2
(a(81/16) - a(27/8)) = 27/2
(81a/16 - 27a/8) = 27/2

To simplify this equation, let's work with the denominator:

81a/16 - 27a/8 = 27/2
(81a - 54a)/16 = 27/2
27a/16 = 27/2

To find a, let's cross-multiply:

27a = 16 * 27/2
27a = 432/2
27a = 216
a = 8

So, the first term (a) is 8 and the common ratio (r) is 3/2. Keep smiling, because we just solved our exponential puzzle!

To find the first term and the common ratio of an exponential sequence, we need to set up a system of equations using the given information.

Let's assume that the first term of the sequence is "a" and the common ratio is "r".

The formula for the n-th term of an exponential sequence is given by:

an = a * r^(n-1)

Now, let's use the given information to set up the equations:

1) The fifth term is greater than the fourth term by 27/2:

a * r^4 = a * r^3 + 27/2

2) The fourth term is greater than the third term by 9:

a * r^3 = a * r^2 + 9

Now, we have a system of two equations that we can solve to find the values of "a" and "r".

To solve the system, we can use algebraic methods such as substitution or elimination. In this case, let's use substitution.

From equation 1), we can isolate a * r^4 by subtracting a * r^3 from both sides:

a * r^4 - a * r^3 = 27/2

Factoring out "a" on the left side:

a * (r^4 - r^3) = 27/2

Now, divide both sides of the equation by (r^4 - r^3):

a = (27/2) / (r^4 - r^3)

We have obtained the value of "a".

Now, substitute this value of "a" into equation 2):

a * r^3 = a * r^2 + 9

[(27/2) / (r^4 - r^3)] * r^3 = [(27/2) / (r^4 - r^3)] * r^2 + 9

Simplify both sides of the equation:

(27/2) * r = (27/2) * r^2 + 9 * (r^4 - r^3)

Expand and rearrange the equation:

(27/2) * r^2 + 9 * r^3 - 9 * r^4 = 0

This equation is a fourth-degree polynomial in terms of "r". You can solve it using various methods such as factoring, synthetic division, or numerical methods like graphing or using a calculator.

Once you obtain the values of "r", substitute them back into equation 1) to find the corresponding values of "a".