If m, n are natural numbers, m > n, sum of mth and nth term of an increasing AP is 2m and their product is
m^2–- n^2, then what is the (m+n)th term of the A.P.?
The first term of a GP is -3and the square of the term is equal to its 4th term . Find the 7th term
To find the (m+n)th term of the arithmetic progression (AP), we need to first determine the values of m and n.
From the given information, we know that the sum of the mth and nth terms of the AP is 2m and their product is m^2 - n^2.
First, let's express the mth and nth terms of the AP using the general formulas:
mth term = a + (m - 1)d
nth term = a + (n - 1)d
where a is the first term of the AP and d is the common difference.
We're given that the sum of the mth and nth terms is 2m, so we can set up the equation:
(a + (m - 1)d) + (a + (n - 1)d) = 2m
Simplifying this equation gives us:
2a + (m + n - 2)d = 2m
Next, we know that the product of the mth and nth terms is m^2 - n^2. So, we can set up another equation:
(a + (m - 1)d) * (a + (n - 1)d) = m^2 - n^2
Expanding and simplifying this equation yields:
a^2 + (m + n - 1)ad + (mn - m - n + 1)d^2 = m^2 - n^2
Now, we have a system of two equations:
2a + (m + n - 2)d = 2m ----------------- (Equation 1)
a^2 + (m + n - 1)ad + (mn - m - n + 1)d^2 = m^2 - n^2 ------ (Equation 2)
By solving these equations simultaneously, we can determine the values of a and d.
Once we have values for a and d, we can find the (m + n)th term using the general formula:
(m + n)th term = a + ((m + n) - 1)d
Plug in the values of a and d into this formula to find the (m + n)th term of the AP.