The 21st, 37th and 56th terms of an AP are consecutive terms of a GP. Find the common ratio of the GP.
just use what you know about this stuff
(a+36d)/(a+20d) = (a+55d)/(a+36d)
(a+36d)^2 = (a+55d)(a+20d)
a^2+72ad+1296d^2 = a^2+75ad+1100d^2
3ad = 196d^2
3a = 196d
That is, for any value of n,
a=196n
d=3n
So, there is no unique solution.
If n=1, then a=196 and d=3. The terms are
196+20*3 = 256
196+36*3 = 304
196+55*3 = 361
304/256 = 361/304
You can easily verify that it works for any value of n.
To find the common ratio of the geometric progression (GP), we can use the formula for the nth term of an arithmetic progression (AP).
The formula for the nth term of an AP is:
an = a + (n-1)d,
where an is the nth term, a is the first term, n is the position of the term, and d is the common difference.
Given that the 21st, 37th, and 56th terms of the AP are consecutive terms of a GP, let's denote the first term of the AP as a, and the common difference as d.
We know that the difference between the 21st and 37th terms is equal to the difference between the 37th and 56th terms. Mathematically, this can be expressed as:
(a + 20d) - (a + 36d) = (a + 36d) - (a + 55d).
Simplifying this equation gives:
20d - 36d = 36d - 55d,
-16d = -19d.
Dividing both sides by -d (assuming d ≠ 0) gives:
16 = 19.
However, this equation is not possible to satisfy unless we made some error in our calculations or assumptions. Therefore, we cannot find the common ratio of the GP using the given information. Please double-check the question or provide additional details if necessary.