If a1, a2, a10 are in an AP, and a5 = 27, find the sum of the first 10 terms.

Let the arithmetic progression be denoted by {a_n}. We are given that a_5 = 27. The general formula for the nth term of an arithmetic progression is a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

We have a_5 = a_1 + (5-1)d = a_1 + 4d = 27.

The sum of the first n terms of an arithmetic progression is given by S_n = n/2(2a_1 + (n-1)d).
We want to find S_10, the sum of the first 10 terms. Therefore, we need to find a_1 and d.

We are given that a_1, a_2, a_{10} are in an AP. This doesn't provide any additional information beyond what we already know from the fact that a_5 = 27.

However, we can express a_10 in terms of a_1 and d: a_{10} = a_1 + 9d.
Since a_5 = 27, we have a_1 + 4d = 27.

To find S_10, we use the formula:
S_{10} = 10/2 (2a_1 + (10-1)d) = 5(2a_1 + 9d) = 10a_1 + 45d.

We have one equation with two unknowns: a_1 + 4d = 27. We cannot uniquely determine a_1 and d with only this information. The problem statement is incomplete; it needs another equation or piece of information to solve for S_10. For example, we could be given the value of a_10, a_1, or the common difference, d.