three consecutive terms of a geometric progression are the 1st,2nd and 6th terms of an ap.find the common ratio
first
a = a
a r = a + d
a r^2 = a + 5 d
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a (r-1) = d so a =d/(r-1)
d r^2 / (r-1) = d/(r-1) + 5 d
d r^2 = d + 5d (r-1)
r^2 = 1 + 5 r - 5
r^2 - 5 r +4 = 0
(r - 4)(r-1) = 0
r = 1 is not much of a series so
r = 4
Let's denote the first term of the geometric progression as 'a' and the common ratio as 'r'.
The first term of the AP is given as the 1st term of the geometric progression, which is 'a'.
The second term of the AP is given as the 2nd term of the geometric progression, which is 'ar'.
The sixth term of the AP is given as the 6th term of the geometric progression, which is 'ar^5'.
Now, we can set up an equation using the given information:
The 2nd term of the AP = The 1st term of the AP + (common difference × 1)
=> ar = a + d, but d is unknown.
The 6th term of the AP = The 1st term of the AP + (common difference × 5)
=> ar^5 = a + 5d.
To eliminate the common difference 'd', we'll subtract the first equation from the second equation:
(ar^5) - (ar) = (a + 5d) - (a + d)
ar^5 - ar = 5d - d
(ar^5 - ar) = 4d
Factoring out 'd', we get:
d(r^5 - 1) = 4d
Since 'd' cannot be zero, we can divide both sides of the equation by 'd':
r^5 - 1 = 4
Next, we'll solve this equation:
r^5 = 4 + 1
r^5 = 5
To find the value of 'r', we'll take the fifth root of both sides:
r = ∛5
Therefore, the common ratio of the geometric progression is ∛5.
To find the common ratio of a geometric progression when the terms are also part of an arithmetic progression, you need to use the following steps:
Step 1: Identify the terms given in the problem.
In this case, the 1st, 2nd, and 6th terms of the geometric progression are also the terms of an arithmetic progression.
Let's denote the 1st term of the geometric progression as "a," the common ratio as "r," and the common difference of the arithmetic progression as "d."
So, we have:
First Term of Geometric Progression (1st AP term) = a
Second Term of Geometric Progression (2nd AP term) = ar
Sixth Term of Geometric Progression (6th AP term) = ar^5
Step 2: Express the terms in terms of the corresponding arithmetic progression terms.
For the first term of the geometric progression (1st AP term), it coincides with the 1st term of the arithmetic progression (1st AP term). Therefore, a = a.
For the second term of the geometric progression (2nd AP term), it coincides with the 2nd term of the arithmetic progression (2nd AP term). Therefore, ar = a + d.
And for the sixth term of the geometric progression (6th AP term), it coincides with the 6th term of the arithmetic progression (6th AP term). Therefore, ar^5 = a + 5d.
Step 3: Solve the equations.
From the second equation, we can express "a" in terms of "r" and "d":
ar = a + d
ar - a = d
a(r - 1) = d
a = d / (r - 1)
Now, substitute this expression for "a" in the third equation:
ar^5 = a + 5d
(r^5)(d / (r - 1)) = d / (r - 1) + 5d
r^5 = 1 + 5(r - 1)
r^5 = 1 + 5r - 5
r^5 - 5r + 4 = 0
This is a 5th degree polynomial equation in "r." To find its solutions, you can use numerical methods, such as Newton's method or bisection method, or you can use a computer program or calculator capable of solving polynomial equations.
By finding the solutions to the polynomial equation r^5 - 5r + 4 = 0, you can determine the common ratios for which the given terms form an arithmetic progression.