In an arithmetical progression,the sum of the first five terms is 30,and the third term is equal to the sum of the first two.write down the first five terms of the progression.

To solve this problem and find the first five terms of the arithmetic progression, we need to first understand the properties of an arithmetic progression.

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. We can denote the first term as "a" and the common difference as "d".

Let's break down the given information:
1. The sum of the first five terms is 30.
2. The third term is equal to the sum of the first two terms.

Using this information, we can proceed with solving the problem.

Step 1: Finding the common difference (d)
To find the common difference (d), we can use the formula for the sum of an arithmetic progression:

Sum of n terms (Sn) = (n/2)(2a + (n-1)d)

Given that the sum of the first five terms is 30 (Sn = 30) and the number of terms (n) is 5, we can substitute the values into the formula:

30 = (5/2)(2a + (5-1)d) [Equation 1]

Step 2: Finding the third term (a3)
From the given information, we have that the third term (a3) is equal to the sum of the first two terms:

a3 = a1 + a2 [Equation 2]

In an arithmetic progression, a2 = a1 + d. Substituting this into Equation 2:

a3 = a1 + (a1 + d)
a3 = 2a1 + d [Equation 3]

Step 3: Solving the equations simultaneously
We now have two equations (Equations 1 and 3) with two unknowns (a1 and d). Let's solve these equations simultaneously to find their values.

Substitute Equation 3 into Equation 1:

30 = (5/2)(2a + (5-1)d)
30 = (5/2)(2(2a1 + d) + (5-1)d)
30 = (5/2)(4a1 + 2d + 4d)
30 = 5(2a1 + 6d)
6 = 2a1 + 6d [Equation 4]

We now have a system of linear equations:

2a1 + 6d = 6 [Equation 4]
2a1 + d = a3 [Equation 3]

Solving this system of equations will give us the values of a1 and d.

Subtract Equation 3 from Equation 4:

5d = 6 - a3

We don't have the value of a3, so we can't solve this system of linear equations and find the values of a1 and d. The information given in the problem is insufficient to determine the first five terms of the arithmetic progression.

a+2d = a + a+d

5/2 (2a+4d) = 30

Now just crank it out.