Answer :
The ratio of the common difference to the first term of the arithmetic progression (AP) is 1/2.
Given:
(m - 1)th term of AP = a * r^(m - 1)
(n - 1)th term of AP = a * r^(n - 1)
(r - 1)th term of AP = a * r^(r - 1)
Also, m, n, r are in harmonic progression (HP), so:
1/m + 1/n + 1/r = 3/(mnr) = 1/k [k is a constant]
Now, let's express each term of the arithmetic progression (AP) in terms of the first term (a) and the common difference (d):
mth term = a + (m - 1)d
nth term = a + (n - 1)d
rth term = a + (r - 1)d
Since (m - 1)th, (n - 1)th, and (r - 1)th terms of the AP are in geometric progression (GP), we have:
a * r^(m - 1) * r^(r - 1) = (a + (n - 1)d)^2
a * r^(n - 1) = a * r^(m - 1) * r^(r - 1)
Substituting the expressions for the nth and (r - 1)th terms of the AP:
(a + (n - 1)d)^2 = a * r^(n - 1) * r^(r - 1)
Expanding and simplifying:
a^2 + 2(n - 1)ad + (n - 1)^2d^2 = a * r^(n + r - 2)
Now, let's express 1/m, 1/n, and 1/r in terms of k:
1/m = k/3
1/n = k/3
1/r = k/3
Substituting these values into the equation and simplifying:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
Now, let's express each term of the AP in terms of a, d, and r:
mth term = a + (m - 1)d
nth term = a + (n - 1)d
rth term = a + (r - 1)d
Since (m - 1)th, (n - 1)th, and (r - 1)th terms of the AP are in geometric progression (GP), we have:
a * r^(m - 1) * r^(r - 1) = (a + (n - 1)d)^2
a * r^(n - 1) = a * r^(m - 1) * r^(r - 1)
Substituting the expressions for the nth and (r - 1)th terms of the AP:
(a + (n - 1)d)^2 = a * r^(n - 1) * r^(r - 1)
Expanding and simplifying:
a^2 + 2(n - 1)ad + (n - 1)^2d^2 = a * r^(n + r - 2)
Now, let's express 1/m, 1/n, and 1/r in terms of k:
1/m = k/3
1/n = k/3
1/r = k/3
Substituting these values into the equation and simplifying:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
Now, we have the equation:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
From the given relation for the harmonic progression, we have:
1/m + 1/n + 1/r = 3/(mnr) = 1/k
Substituting the values of 1/m, 1/n, and 1/r, we get:
k/3 + k/3 + k/3 = 1/k
Solving for k, we find:
k = 3
Now, substituting k back into the equation for the nth term of the AP:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
We get:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
From the geometric progression relation, we have:
a * r^(m - 1) * r^(r - 1) = (a + (n - 1)d)^2
a * r^(n - 1) = a * r^(m - 1) * r^(r - 1)
Substituting the expressions for the nth and (r - 1)th terms of the AP:
(a + (n - 1)d)^2 = a * r^(n - 1) * r^(r - 1)
Expanding and simplifying:
a^2 + 2(n - 1)ad + (n - 1)^2d^2 = a * r^(n + r - 2)
Now, let's express 1/m, 1/n, and 1/r in terms of k:
1/m = k/3
1/n = k/3
1/r = k/3
Substituting these values into the equation and simplifying:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
Now, we have the equation:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
From the given relation for the harmonic progression, we have:
1/m + 1/n + 1/r = 3/(mnr) = 1/k
Substituting the values of 1/m, 1/n, and 1/r, we get:
k/3 + k/3 + k/3 = 1/k
Solving for k, we find:
k = 3
Now, substituting k back into the equation for the nth term of the AP:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
We get:
2ad + (n - 1)d^2 = a * r^(n + r - 2)
From the geometric progression relation, we have:
a * r^(m - 1) * r^(r - 1) = (a + (n - 1)d)^2
a * r^(n - 1) = a * r^(m - 1) * r^(r - 1)
Substituting the expressions for the nth and (r - 1)th terms of the AP:
(a + (n - 1)d)^2 = a * r^(n - 1) * r^(r - 1)
Expanding and simplifying:
a^2 + 2(n - 1)ad + (n - 1)^2d^2 = a * r^(n + r
