The sum of terms of an A.P is 50 it's first and last terms are 7and 3 respectively. Find the number of the terms A.P

Answer 1

Let the number of terms in the A.P be n.
The sum of n terms of an A.P is given by the formula: Sn = n/2 [2a + (n-1)d], where a is the first term, d is the common difference, and Sn is the sum of n terms.

Given that the sum of the terms of the A.P is 50, we have:
50 = n/2 [2(7) + (n-1)d]
50 = n/2 [14 + (n-1)d]
50 = n/2 [14 + 3(n-1)]

Now, we know that the first term a = 7 and the last term l = 3. Using the formula for the nth term of an A.P: ln = a + (n-1)d, we get:
3 = 7 + (n-1)d
-4 = (n-1)d
-4 = nd - d
d = -4/(n-1)

Substitute d into the equation 50 = n/2 [14 + 3(n-1)], we get:
50 = n/2 [14 + 3(n-1)]
50 = n/2 [14 + 3n - 3]
50 = n/2 (11 + 3n)
100 = n (11 + 3n)
100 = 11n + 3n^2
3n^2 + 11n - 100 = 0

Solving the quadratic equation, we get n = 4 or n = -8/3. Since the number of terms cannot be negative, the number of terms in the A.P is 4.