The sum of the first two term of a g.p is x and the sum is of the last two terms is y. If there are n terms in all. Find the common ratio
a + ar = x
ar^(n-2) + ar^(n-1) = y
a(1+r) = x
ar^(n-2) (1+r) = y
xr^(n-2) = y
r^(n-2) = y/x
r = (y/x)^(1/(n-2))
Well, isn't math just a series of mental gymnastics? To find the common ratio, let's call it 'r', we need to work our way through this riddle.
The sum of the first two terms in the geometric progression is x. So, let's write that as:
a + ar = x
Now, the sum of the last two terms is y. That means the last term would be 'a' times 'r' to the power of n minus 1, and the second-to-last term would be 'a' times 'r' to the power of n minus 2. Let's put that into an expression:
a times 'r' to the power of n minus 2 + a times 'r' to the power of n minus 1 = y
All right, now we have a system of two equations! It's like trying to untangle a clown's wig! Let's solve this puzzle and find 'r'.
From the first equation, we can rewrite it as:
a(1 + r) = x
Now, let's divide the second equation by 'a' to make it easier to work with:
r to the power of n minus 2 + r to the power of n minus 1 = y/a
Okay, we're getting closer to the punchline! Now, let's simplify both equations:
1 + r = x/a
And
r to the power of n minus 2 + r to the power of n minus 1 = y/a
Oh, what a tangled web of variables we have here! But fear not, dear friend, we're almost done. We can simplify further by subtracting 1 from both sides in the first equation:
r = (x/a) - 1
Now, let's substitute this value of 'r' back into the second equation and see what we get:
((x/a) - 1) to the power of n minus 2 + ((x/a) - 1) to the power of n minus 1 = y/a
And there you have it! The common ratio, 'r', is equal to (x/a) - 1. Of course, the answer is never quite as simple as it seems. But don't worry, solving math problems is just a big, never-ending circus!
To find the common ratio in this geometric progression (G.P.), we'll use the given information on the sum of the first two terms (x) and the sum of the last two terms (y).
Let's assume the common ratio of the G.P. is denoted by 'r'.
The formula for the sum of the first n terms of a G.P. is: S = a * (r^n - 1) / (r - 1), where 'a' is the first term and 'S' is the sum.
Given that the sum of the first two terms is x, we have:
x = a * (r^2 - 1) / (r - 1) ------ Equation 1
Similarly, the sum of the last two terms is y, so we have:
y = a * (r^n - r^(n-1)) / (r - 1) ------ Equation 2
To find the common ratio (r), we can use the equations (1) and (2).
First, simplify equation (1):
x * (r - 1) = a * (r^2 - 1)
xr - x = a * (r^2 - 1)
Next, simplify equation (2):
y * (r - 1) = a * (r^n - r^(n-1))
yr - y = a * (r^n - r^(n-1))
Now, let's divide equation (2) by equation (1) to eliminate 'a':
(yr - y) / (xr - x) = (r^n - r^(n-1)) / (r^2 - 1)
Simplifying further:
(yr - y) / (xr - x) = (r^(n-1) * (r - 1)) / ((r + 1) * (r - 1))
Canceling out (r - 1):
(yr - y) / (xr - x) = (r^(n-1)) / (r + 1)
Cross-multiplying:
r * (yr - y) = x * (r^(n-1))
Expanding:
ryr - ry = xr^(n-1)
Rearranging terms:
ryr - xr^(n-1) = ry
Simplifying further:
r * (yr - xr^(n-1)) = ry
Dividing both sides by r (assuming r ≠ 0):
yr - xr^(n-1) = y
Finally, we can rearrange the equation in terms of the common ratio 'r':
xr^(n-1) = yr - y
xr^(n-1) = y(r - 1)
Dividing both sides by x (assuming x ≠ 0):
r^(n-1) = y(r - 1) / x
Taking the n-1-th root of both sides:
r = (y(r - 1) / x)^(1/(n-1))
So, the common ratio 'r' in the G.P. is given by the formula:
r = (y(r - 1) / x)^(1/(n-1))
Please note that the common ratio could have multiple values depending on the given values of 'x', 'y', and 'n'.
To find the common ratio (r) of a geometric progression (g.p), we can use the formula for the sum of the first two terms and the sum of the last two terms. Let's break down the problem step by step:
1. The sum of the first two terms of a g.p is x.
We know that the sum of the first two terms in a g.p is given by the formula:
S2 = a + ar,
where S2 is the sum of the first two terms, a is the first term, and r is the common ratio.
Given that S2 = x, we can rewrite the formula as:
x = a + ar.
2. The sum of the last two terms of a g.p is y.
The sum of the last two terms in a g.p can be calculated using the formula:
Sn = a * (r^n - 1) / (r - 1),
where Sn is the sum of the n terms, a is the first term, r is the common ratio, and n is the number of terms.
Since we have n terms in total, the sum of the last two terms would be:
y = a * (r^n - 1) / (r - 1).
Now, we have two equations derived from the given information:
x = a + ar,
y = a * (r^n - 1) / (r - 1).
To find the common ratio (r), we can solve these equations simultaneously.
1. Rearrange the first equation to express a in terms of x:
x = a + ar.
x - ar = a.
a = x / (1 + r).
2. Substitute the value of a in the second equation with x / (1 + r):
y = a * (r^n - 1) / (r - 1).
y = (x / (1 + r)) * (r^n - 1) / (r - 1).
Now, we have a single equation involving the common ratio (r) and the given variables (x, y, and n):
y = (x / (1 + r)) * (r^n - 1) / (r - 1).
Simplifying and solving this equation for r might involve using algebraic techniques or numerical methods such as graphing or iterative approximation. Unfortunately, there is no direct formula to find the common ratio in terms of x, y, and n in this scenario.
Therefore, you will need to apply numerical methods or algebraic techniques to solve the equation and find the value of the common ratio (r) based on the given values of x, y, and n.