The fifth term of a Gp is 18 morethan the second term. If the difference between the seventh and the fourth term is 162. Find the first term, the common ratio and the tenth term
a=_60
r=3
T10=118980
To find the first term, common ratio, and tenth term of a geometric progression (GP), we can use the given information to set up a system of equations.
Let's start with finding the first term (a) and common ratio (r) of the GP:
1. The fifth term of a GP is 18 more than the second term:
This can be written as:
a * r^4 = a + 18 ... (equation 1, as the fifth term is a * r^4 and the second term is a)
2. The difference between the seventh and fourth term is 162:
This can be written as:
a * r^6 - a * r^3 = 162 ... (equation 2, as the seventh term is a * r^6 and the fourth term is a * r^3)
Now we have two equations with two unknowns (a and r). Let's solve them simultaneously:
From equation 1:
a * r^4 = a + 18
Divide both sides by a:
r^4 = 1 + 18/a
From equation 2:
a * r^6 - a * r^3 = 162
Divide both sides by a:
r^6 - r^3 = 162/a
Now, we can substitute the value of (r^4) from equation 1 into equation 2:
(1 + 18/a)^3 - (1 + 18/a) = 162/a
Simplifying this equation will give us the value of a.
Once we have the value of a, we can substitute it back into equation 1 to find the value of r:
a * r^4 = a + 18
Finally, to find the tenth term, we can use the formula for the nth term of a GP:
nth term = a * r^(n-1)
Substituting n = 10, a, and r into this formula will give us the tenth term.
Please note that solving the final equation for the value of a and r might require numerical methods or trial and error as it won't always have a straightforward algebraic solution.
ar^4 = ar+18
ar^6 - ar^3 = 162
Now just solve for a and r, and find ar^9