The fifth of an exponential sequence (GP) is greater than the fourth term by 13.5 and fourth term is greater than the thrid by 9.Find:common ratio and the common difference
Let the first term of the exponential sequence be a, and let the common ratio be r.
We are given that the fifth term is greater than the fourth term by 13.5, so we can set up the equation:
ar^4 = ar^3 + 13.5
We are 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 these equations, we can divide the first equation by the second equation:
(ar^4) / (ar^3) = (ar^3 + 13.5) / (ar^2 + 9)
r = (ar^3 + 13.5) / (ar^2 + 9)
Multiplying both sides of the equation by (ar^2 + 9) gives us:
r(ar^2 + 9) = ar^3 + 13.5
Expanding both sides of the equation:
ar^3 + 9r = ar^3 + 13.5
Subtracting ar^3 from both sides of the equation:
9r = 13.5
Dividing both sides of the equation by 9 gives us:
r = 13.5 / 9
r = 1.5
Now we can substitute r = 1.5 into one of the equations to find the common difference. Let's use the equation ar^3 = ar^2 + 9:
a(1.5)^3 = a(1.5)^2 + 9
a(3.375) = a(2.25) + 9
3.375a = 2.25a + 9
1.125a = 9
a = 9 / 1.125
a = 8
Therefore, the common ratio is 1.5 and the common difference is 8.