The sum of 8 term of an a-p is 100 while the sum of 20 term is 880 find the sum of 12 term
To find the sum of the 12th term of an arithmetic progression (a-p), we need to use the given information.
In an arithmetic progression, each term is obtained by adding a common difference (d) to the previous term. Let's denote the first term as 'a' and the common difference as 'd'.
We know that the sum of the first 8 terms is 100. The formula to calculate the sum of the first n terms of an arithmetic progression is:
Sn = (n/2)(2a + (n-1)d)
Using the given information, we can substitute the values into this formula:
100 = (8/2)(2a + (8-1)d)
100 = 4(2a + 7d)
100 = 8a + 28d - Equation 1
We are also given that the sum of the first 20 terms is 880:
880 = (20/2)(2a + (20-1)d)
880 = 10(2a + 19d)
88 = 2a + 19d - Equation 2
Now, we have a system of two equations with two variables (a and d). We can solve this system of equations to find the values of 'a' and 'd'.
To solve the system, subtract Equation 2 from Equation 1:
100 - 88 = 8a + 28d - (2a + 19d)
12 = 6a + 9d
Simplifying further, we get:
2 = a + (3/2)d - Equation 3
Now, substitute the value of 'a' from Equation 3 into Equation 2:
88 = 2(a + (3/2)d) + 19d
88 = 2a + 3d + 19d
88 = 2a + 22d
44 = a + 11d - Equation 4
Now, we have a new equation that relates 'a' and 'd'.
To find the sum of the 12th term, we need to calculate the value of the 12th term (a12). The formula to calculate the nth term of an arithmetic progression is:
an = a + (n-1)d
Substituting n=12, we get:
a12 = a + 11d
Since we have the values of 'a' and 'd' from Equation 4, we can substitute them into the formula to find the 12th term:
a12 = 44 + 11d
Finally, to find the sum of the 12th term (S12), we can use the formula for the sum of the first n terms again:
S12 = (12/2)(2a + (12-1)d)
Substituting the value of 'a12' into this formula, we get:
S12 = (12/2)(2(44) + (12-1)d)
S12 = 6(88 + 11d)
S12 = 528 + 66d
Therefore, the sum of the 12th term is given by the expression 528 + 66d. To find the specific value, we need to know the value of 'd'.
sum(8) = 4(2a + 7d) = 100
2a + 7d = 25 ***
sum(20) = 10(2a + 19d) = 880
2a + 19d = 88 ***
subtract them:
12d = 63
d = 21/4
in ***
2a + 7(21/4) = 25
2a = -47/4
a = -47/8
sum12 = 6(2(-47/8) + 11(21/4) = 276