In a decreasing gp the sum of the second and third terms is 36 and the difference between these two terms is 18.find a)the first term b)the nth term
ar + ar^2 = 36
ar - ar^2 = 18
dividing,
r(1+r)/(r(1-r)) = 2
1+r = 2(1-r)
r = 1/3
a = 81
The GP is 81,27,9,3,...
right
please show solution
To find the first term (a) and the nth term in a decreasing geometric progression (GP), we need to use the given information.
Let's start by setting up the GP formula for the terms:
a, ar, ar^2, ar^3, ...
Given information:
In a decreasing GP, the sum of the second and third terms is 36: ar + ar^2 = 36, which can be written as ar(1 + r) = 36.
The difference between these two terms is 18: ar^2 - ar = 18.
Now, we can solve these two equations simultaneously to find the values of 'a' and 'r'.
Equation 1: ar(1 + r) = 36
Equation 2: ar^2 - ar = 18
First, let's simplify Equation 2 by dividing both sides by 'a':
r^2 - r = 18/a
Now, let's substitute this value into Equation 1:
a(r^2 - r)(1 + r) = 36
(r^2 - r)(1 + r) = 36/a
Expanding and rearranging this equation, we get:
r^3 + r^2 - r^2 - r = 36/a
r^3 - r = 36/a
Now, let's simplify it further:
r(r^2 - 1) = 36/a
r(r - 1)(r + 1) = 36/a
Since r - 1 ≠ 0 (assumption that GP is non-zero), we can divide both sides by r - 1:
r(r + 1) = 36/a
Next, substituting the value of ar from the first equation into the second equation, we get:
ar + a = 36/a
Simplifying this equation, we have:
a(r + 1) = 36/a
Now, let's solve these two equations to find the values of 'a' and 'r':
a(r + 1) = 36/a
a^2(r + 1) = 36
From this equation, we can see that r must be -1 (since if r + 1 ≠ 0, then a^2 = 36, but a^2 cannot be negative).
So, when r = -1, substituting this value into the first equation:
a(-1 + 1) = 36/a
0 = 36/a
Since a cannot be zero and 0 = 36/a implies 36 = 0, there is no valid solution for a and r when r = -1.
Therefore, there are no valid values for both the first term (a) and the nth term in this given situation.