The sum of 18 terms of an A.P is 504 while the sum of its 9 terms is -126.Find the sum of its 30 term
I will assume you know the formulas for the sum of n terms of an AP
Sum(18) = 9(2a + 17d) = 504
2a + 17d = 56
Sum(9) = (9/2)(2a + 8d) = -126
2a + 8d = -28
subtract them: 9d =84
d = 28/3
sub into 2a+8d=-28
2a + 224/3 = -28
a = -154/3
sum(30) = 15(2a + 29d) = .....
To find the sum of the 30th term of an arithmetic progression (A.P.), we need to know the first term and the common difference.
Given Information:
Sum of 18 terms of the A.P. = 504 ---> 1st equation
Sum of 9 terms of the A.P. = -126 ---> 2nd equation
To solve this problem, we can use the formula for the sum of an A.P.:
Sum of n terms of an A.P. = (n/2)[2a + (n-1)d]
Where:
n = number of terms in the A.P.
a = first term of the A.P.
d = common difference between the terms
Let's proceed with finding the values of a (first term) and d (common difference).
1. Finding the value of a (first term):
Using the formula for the sum of 9 terms:
-126 = (9/2)[2a + (9-1)d]
Simplifying this equation, we get:
-126 = 9a + 36d ---> 3rd equation
2. Finding the value of d (common difference):
Using the formula for the sum of 18 terms:
504 = (18/2)[2a + (18-1)d]
Simplifying this equation, we get:
504 = 9a + 171d ---> 4th equation
Now, we have a system of linear equations (3rd equation and 4th equation) with two variables (a and d). We can solve these equations simultaneously to find the values of a and d.
Subtracting equation 4 from equation 3, we eliminate a, and we get:
-630 = -135d
Dividing both sides by -135, we find:
d = 630/135
d = 14/3
Substituting the value of d in equation 3, we find:
-126 = 9a + 36(14/3)
-126 = 9a + 168
-126 - 168 = 9a
-294 = 9a
a = -294/9
a = -98/3
Now that we have found the values of a (first term) and d (common difference), we can find the sum of the 30th term of the A.P. using the sum formula.
Sum of 30 terms of an A.P. = (30/2)[2(-98/3) + (30-1)(14/3)]
Simplifying this equation will give us the sum of the 30th term.
Please calculate the equation above to find the sum of the 30th term of the A.P.