The fifth term of an A.p. Exceeds the second term by 1.The tenth term exceeds twice the fourth term by 3
* find first term and common difference
* sum of first twenty-five terms.
cd = 1 / (5 - 2) = 1/3
t = f + 6 cd
t = 2f + 3 = f + 2
... 4th = -1
... 1st = -1 - (3 * 1/3) = -2
... 25th = -2 + (24 / 3) = 6
s = (-2 + 6) * (25/2)
To find the first term and common difference of an arithmetic progression (A.P.), we can solve a system of equations using the given conditions. Let's denote the first term as 'a' and the common difference as 'd'.
Given:
- The fifth term exceeds the second term by 1: a + 4d = (a + d) + 1
Simplifying the equation:
a + 4d = a + d + 1
3d = 1
d = 1/3
Now, we have the common difference as d = 1/3.
To find the first term, we'll use the second given condition:
- The tenth term exceeds twice the fourth term by 3: a + 9d = 2(a + 3d) + 3
Simplifying the equation:
a + 9d = 2a + 6d + 3
3d = -3
d = -1
Now, we have a different value for the common difference (d = -1). This means the given conditions are contradictory, and there is no solution for this situation.
However, let's assume that the first given condition (difference between the fifth and second terms) is correct. In that case, we can continue solving the problem with d = 1/3.
To find the sum of the first twenty-five terms of an A.P., we can use the formula:
S = (n/2)(2a + (n-1)d)
Here, n = 25 (the number of terms).
Substituting the values we know, a = first term, d = common difference, and n = 25, into the formula:
S = (25/2)(2a + 24(1/3))
Simplifying the equation (multiplying through by 2 to clear the fraction):
2S = 25(2a + 8)
2S = 50a + 200
S = 25a + 100
Thus, the sum of the first twenty-five terms is given by S = 25a + 100.
Please note that without a clear and consistent set of conditions, it is not possible to provide an exact solution to the problem.