1st, 7th and 25th terms of an AP are in three consecutive terms of GP. The 20th term of an AP is 22. Find
Common difference of an AP
The 10th of the GP
Let's call the first term of the AP "a" and the common difference of the AP "d".
Given that the 1st, 7th, and 25th terms of the AP are in three consecutive terms of a GP, we can write the following equations:
a + (6d) = ar
a + (24d) = ar^2
Simplifying these equations, we have:
1) ar - a = 6d
2) ar^2 - a = 24d
Since the 20th term of the AP is 22, we can write another equation:
a + (19d) = 22
From equation 1), we have:
ar = 6d + a
r = (6d + a)/a
Substituting r in equation 2), we have:
(6d + a)r^2 - a = 24d
(6d + a)((6d + a)^2/a^2) - a = 24d
(6d + a)(36d^2 + 12ad + a^2)/a^2 - a = 24d
Expanding and simplifying further, we have:
216d^3/a^2 + 96d^2 + 42d + a = 24d
Rearranging the terms, we have:
216d^3/a^2 + 96d^2 + 18d - 22d + a = 0
Since the 20th term of the AP is 22, we also have:
a + 19d = 22
We have a system of equations here that we can solve to find the values of a and d. However, finding the common difference (10th term of the GP) is not possible with the provided information.
To find the common difference of an arithmetic progression (AP), we can use the formula:
nth term = a + (n-1)d
where "a" is the first term, "n" is the term number, and "d" is the common difference.
Given that the 20th term of the AP is 22, we can set up the equation:
22 = a + (20-1)d
22 = a + 19d --(1)
Now, let's consider the terms of the AP that are also in a geometric progression (GP). We are told that the 1st, 7th, and 25th terms of the AP are in three consecutive terms of the GP.
Let's assume the terms of the GP are b, br, and br^2, where "b" is the first term and "r" is the common ratio. Therefore, we can form the following equations:
a = b
a + 6d = br
a + 24d = br^2
Now, let's solve the system of equations to find the values of "a" and "d".
From the above equations, we can rewrite them in terms of "a" and "d":
b = a
br = a + 6d
br^2 = a + 24d
Simplifying the second equation, we have:
br = a + 6d
b = (a + 6d)/r --(2)
Substituting equation (2) into equation (3), we get:
(a + 6d)/r^2 = a + 24d
a + 6d = ar^2 + 24dr^2
a(1 - r^2) = 18dr^2
a = (18dr^2)/(1 - r^2) --(4)
Substituting equation (4) into equation (1), we have:
22 = (18dr^2)/(1 - r^2) + 19d
Now, we can solve this equation for "d".
Simplifying the equation:
22(1 - r^2) = 18dr^2 + 19d(1 - r^2)
22 - 22r^2 = 18dr^2 + 19d - 19dr^2
Rearranging the terms:
22r^2 + 19dr^2 - 18dr^ - 22 +19d = 0
Factoring out "r^2" and "d", we have:
r^2(22 + 19d) - r(18d - 19) - (22 - 19d) = 0
To find the values of "r" and "d", we need to solve this quadratic equation. However, without any additional information or specific values, we cannot determine the exact values.
Therefore, we cannot find the common difference of the AP or the 10th term of the GP unless we have more information or specific values.
To find the common difference of the arithmetic progression (AP), we can use the formula for the nth term of an AP:
an = a1 + (n-1)d
Where:
an = nth term of the AP
a1 = first term of the AP
d = common difference of the AP
n = position of the term in the AP
Given that the 20th term of the AP is 22, we can substitute these values into the formula:
22 = a1 + (20-1)d
Simplifying the equation, we have:
22 = a1 + 19d
Now, let's look at the given information: the 1st, 7th, and 25th terms of the AP are in three consecutive terms of a geometric progression (GP). In a geometric progression, each term is found by multiplying the previous term by a constant ratio (r).
Let's assume that the common ratio of the GP is r.
So, the three consecutive terms of the GP that contain the 1st, 7th, and 25th terms of the AP would be ar^6, ar^24, and ar^96.
We know that these three terms are also terms of the AP. Therefore, we can set up the following equations:
a1 + 6d = ar^6
a1 + 24d = ar^24
a1 + 96d = ar^96
We have three equations with three unknowns: a1, d, and r. We need to solve these equations simultaneously.
One way to proceed is to eliminate a1 from the equations. Let's subtract the first equation from the second equation:
(a1 + 24d) - (a1 + 6d) = ar^24 - ar^6
18d = ar^6(r^18 - 1)
Now, subtract the second equation from the third equation:
(a1 + 96d) - (a1 + 24d) = ar^96 - ar^24
72d = ar^24(r^72 - 1)
Divide these two equations:
(18d) / (72d) = [ar^6(r^18 - 1)] / [ar^24(r^72 - 1)]
1/4 = r^-18 / r^-72
1/4 = r^54
Taking the fourth root of both sides of the equation:
r = (1/4)^(1/54)
Now, let's substitute r back into the second equation:
a1 + 24d = ar^24
a1 + 24d = a(1/4)^(24/54)
a1 + 24d = a(1/4)^(4/9)
a1 + 24d = a^(4/9) / 2^2
So, from the above equation, we can see that (a1 + 24d) must be equal to a perfect square (a^(4/9) / 2^2). Therefore, a1 + 24d must be a multiple of 4.
Now we can use the equation we derived earlier:
a1 + 19d = 22
Since (a1 + 24d) is a multiple of 4, let's substitute it with 4k, where k is a positive integer:
4k = 22 - 19d
4k = 22 - 19d
d = (22-4k)/19
Therefore, the solution for the common difference of the arithmetic progression (AP) is given by the expression (22-4k)/19.
Now, to find the 10th term of the geometric progression (GP), we can use the formula:
tn = a1 * r^(n-1)
Substituting the given values:
t10 = a * (1/4)^(10-1)
t10 = a * (1/4)^9
t10 = a / 4^9
Hence, the 10th term of the geometric progression (GP) is given by the expression a / 4^9.