A.P. with given terms
The given arithmetic progression (A.P.) has the terms -12, -9, -6, and 21.

Number of terms
To find the number of terms in the A.P., we need to count the given terms. The A.P. has 4 terms.

Common difference
To find the common difference (d) of an A.P., we subtract any two consecutive terms. Let's take the first two terms: -9 - (-12) = 3. Therefore, the common difference is 3.

Formula to find the sum of an A.P.
The sum of an A.P. can be found using the formula:

Sn = (n/2)[2a + (n-1)d]

Where:
- Sn is the sum of the first n terms of the A.P.
- a is the first term of the A.P.
- d is the common difference of the A.P.
- n is the number of terms in the A.P.

Sum of the given A.P.
Using the formula, we can find the sum of the given A.P.:

S4 = (4/2)[2(-12) + (4-1)(3)]
= (2)[-24 + 3(3)]
= (2)[-24 + 9]
= (2)[-15]
= -30

Therefore, the sum of the given A.P. is -30.

Adding 1 to each term
To obtain a new A.P. where 1 is added to each term, we simply add 1 to each given term.

The new A.P. terms are:
-12 + 1 = -11
-9 + 1 = -8
-6 + 1 = -5
21 + 1 = 22

Sum of the new A.P.
To find the sum of the new A.P., we can use the same formula as before:

S4' = (4/2)[2(-11) + (4-1)(3)]
= (2)[-22 + 3(3)]
= (2)[-22 + 9]
= (2)[-13]
= -26

Therefore, the sum of the new A.P. is -26.

Conclusion
In conclusion, the given A.P. has 4 terms. When 1 is added to each term, a new A.P. is formed with a sum of -26. The sum of the given A.P. is -30.