the ninth term of an arithmetic progression is 52ind the and the sum of the first twelve terms is 414.find the first term and the common difference.
a + 8d = 52
(12/2)(2a + 11d) = 414
or
2a + 11d = 69
solve these two equations for a and d
d=69&a1=2
ur dumb
To find the first term and the common difference of an arithmetic progression, we can use the formulas:
n-th term of an arithmetic progression (Tn) = a + (n - 1)d
Sum of the first n terms of an arithmetic progression (Sn) = (n/2)(2a + (n-1)d)
Let's use these formulas to solve the given problem.
Given:
The 9th term (T9) = 52
The sum of the first twelve terms (S12) = 414
Step 1: Finding the common difference (d)
We are given:
T9 = a + (9 - 1)d = 52 (Equation 1)
We need to solve Equation 1 to find the common difference (d).
Using the values given, we can rewrite Equation 1:
a + 8d = 52 (Equation 1)
Step 2: Finding the first term (a)
We are also given:
S12 = (12/2)(2a + (12-1)d) = 414 (Equation 2)
We need to solve Equation 2 to find the first term (a), using the value of the common difference (d) obtained from Step 1.
Using the values given, we can rewrite Equation 2:
6(2a + 11d) = 414
12a + 66d = 414 (Equation 2)
Step 3: Solving the system of equations (Equation 1 and Equation 2)
Now, we have a system of two equations with two unknowns (a and d):
a + 8d = 52 (Equation 1)
12a + 66d = 414 (Equation 2)
We can solve this system of equations to find the values of a and d. I will use the method of substitution.
From Equation 1, we can express a in terms of d:
a = 52 - 8d (Equation 3)
Substitute Equation 3 into Equation 2:
12(52 - 8d) + 66d = 414
624 - 96d + 66d = 414
-30d = 414 - 624
-30d = -210
Divide by -30:
d = (-210)/(-30)
d = 7
Step 4: Finding the first term (a)
Substitute the value of d = 7 into Equation 1 to find the first term (a):
a + 8(7) = 52
a + 56 = 52
a = 52 - 56
a = -4
Therefore, the first term (a) is -4 and the common difference (d) is 7.