the 9th term of an A.P is 52 while he 12th term is 70. find the sum of is 20 term

a+8d = 52

a+11d = 70
subtract them
3d = 18
d = 6
back into the first:
a + 48 = 52
a = 4

term20 = a+19d
4 + 19(6) = 118

To find the sum of the first 20 terms of an arithmetic progression (A.P.), we need to determine the first term (a) and the common difference (d).

We are given that the 9th term is 52 and the 12th term is 70.

Let's start by finding the common difference (d). We can do this by subtracting the 9th term from the 12th term:

d = 70 - 52
d = 18

Now, we can find the first term (a) using the formula for the nth term of an A.P.:

a + (n - 1) * d

Using the 9th term to substitute n = 9 and substituting the common difference as 18, we can solve for a:

a + (9 - 1) * 18 = 52
a + 8 * 18 = 52
a + 144 = 52
a = 52 - 144
a = -92

So, the first term (a) is -92 and the common difference (d) is 18.

Now, we can find the sum of the first 20 terms of the A.P. Using the formula for the sum of an A.P.:

S = (n/2) * (2a + (n - 1) * d)

Substituting a = -92, d = 18, and n = 20 into the formula, we get:

S = (20/2) * (2(-92) + (20 - 1) * 18)
S = 10 * (-184 + 19 * 18)
S = 10 * (-184 + 342)
S = 10 * 158
S = 1580

Therefore, the sum of the first 20 terms of the given A.P. is 1580.