In an ascending geometric sequence, (𝑎5)2=3𝑎10, and for any three consecutive terms, 2(𝑎𝑛+𝑎𝑛+2)=5𝑎𝑛+1. Find the first term of the geometric sequence.

To solve this problem, we need to use the properties of geometric sequences and the given equations.

1. In a geometric sequence, each term is the product of the previous term and a constant ratio (r). So, we can express the 5th term (a5) and the 10th term (a10) in terms of the first term (a1) and the ratio (r) as follows: a5 = a1 * r^4 and a10 = a1 * r^9.

2. Substitute these expressions into the first given equation (a5)^2 = 3a10, we get: (a1 * r^4)^2 = 3 * a1 * r^9. Simplify this equation, we get: a1^2 * r^8 = 3 * a1 * r^9. Divide both sides by a1 * r^8, we get: a1 = 3/r.

3. For the second given equation, we know that for any three consecutive terms in a geometric sequence, the middle term is the geometric mean of the other two. So, we can express the equation 2(a_n + a_n+2) = 5a_n+1 as 2(a1 * r^(n-1) + a1 * r^(n+1)) = 5 * a1 * r^n. Simplify this equation, we get: 2(r - r^3) = 5r^2. Divide both sides by 2r, we get: 1 - r^2 = 5/2 * r. This is a quadratic equation in terms of r.

4. Solve this quadratic equation, we get two possible values for r: r = 1/2 or r = -5.

5. Substitute these values into the equation a1 = 3/r, we get two possible values for a1: a1 = 6 or a1 = -3/5.

6. However, since the sequence is ascending, the ratio must be greater than 1. So, the valid solution is a1 = -3/5 and r = -5.

So, the first term of the geometric sequence is -3/5.