The sum of the first 9 terms of an A.P is 72 and the sum of the next 9 terms is 71, find the A.P

I don't know what "A.P" means.

Sure you do. :-)

The question
a) is obviously about an arithmetic progression
b) has already been posted three times!?!?!?!!?

To find the arithmetic progression (AP), we need to determine the first term (a) and the common difference (d).

Let's start by finding the sum of the first 9 terms of the AP. We're given that the sum of these terms is 72.

The formula to calculate the sum of the first n terms of an AP is:
Sn = n/2 * [2a + (n-1)d]

where:
Sn is the sum of the first n terms
a is the first term
d is the common difference
n is the number of terms

Substituting the given values, we have:
72 = 9/2 * [2a + (9-1)d]

Simplifying this equation, we get:
72 = 4.5 * [2a + 8d]
16a + 64d = 32

Similarly, let's find the sum of the next 9 terms. We're given that the sum of these terms is 71.

Using the same formula, we have:
71 = 9/2 * [2(a + 9d) + (9-1)d]

Simplifying this equation, we get:
71 = 4.5 * [2a + 26d]
16a + 130d = 284

Now, we have a system of two linear equations:
16a + 64d = 32
16a + 130d = 284

To solve this system of equations, we can subtract the first equation from the second equation:
(16a + 130d) - (16a + 64d) = 284 - 32
66d = 252
d = 252 / 66
d = 3.8181 (approximately)

Now, we can substitute this value of d into the first equation to find the value of a:
16a + 64(3.8181) = 32
16a + 243.69 = 32
16a = -211.69
a = -211.69 / 16
a = -13.2306 (approximately)

Therefore, the first term (a) of the AP is approximately -13.2306 and the common difference (d) is approximately 3.8181.