the sum of 11 terms of an a.p is 891 find the 29th and 45th terms of the common difference is 15
Let's assume that the first term of the arithmetic progression is 'a'. The sum of 11 terms can be calculated using the formula S11 = (11/2)(2a + (11-1)d), where 'd' is the common difference.
Given S11 = 891, we have:
891 = (11/2)(2a + 10d)
99 = 2a + 10d - Equation 1
We need to find the 29th term and the 45th term. Using the formula for the nth term of an arithmetic progression, we have:
a1 = a + 0d (1st term)
a29 = a + 28d (29th term)
a45 = a + 44d (45th term)
Subtracting Equation 1 from both a29 and a45:
a29 - Equation 1: a + 28d - (2a + 10d) = -a + 18d
a45 - Equation 1: a + 44d - (2a + 10d) = -a + 34d
We can set up a system of equations using the given information:
1) (-a + 18d) + (-a + 34d) = 891 (sum of 11 terms)
2) -a + 18d = a45 - a29 (subtracting Equation 1 from a45 and a29)
Simplifying Equation 1:
-2a + 52d = 891 - Equation 3
Simplifying Equation 2:
-a + 18d = -a + 34d
18d - 34d = a - a
-16d = 0
d = 0
From Equation 3, we can substitute d = 0:
-2a + 52(0) = 891
-2a = 891
a = -891/2
a = -445.5
Thus, the first term 'a' is -445.5 and the common difference 'd' is 0. Since the common difference is 0, this is not an arithmetic progression. Therefore, there are no 29th and 45th terms in the progression.