To insert two arithmetic progressions (APs) between the numbers 2 and 8, we first note the sequence we want. The two APs will be inserted between 2 and 8 as follows:
Let the numbers be represented as: 2, \( a_1 \), \( a_2 \), 8
An arithmetic progression is defined by the property that the difference between consecutive terms is constant.
Let's define the common difference of the first AP as \( d_1 \) and the second AP as \( d_2 \). The first term is 2 and the last term is 8.
- We can start with \( a_1 = 2 + d_1 \) (the first inserted term).
- Then, \( a_2 = 2 + 2d_1 \) (the second inserted term).
- We want the last term, which is 8, so we have: \[ 2 + 2d_1 + d_2 = 8 \] Which simplifies to: \[ 2d_1 + d_2 = 6 \]
To proceed, let's assume \( d_1 = 2 \). Then:
- \( a_1 = 2 + 2 = 4 \)
- \( a_2 = 4 + 2 = 6 \)
So we get the sequence: 2, 4, 6, 8
Thus, the two arithmetic progressions inserted are:
- First AP: 2, 4
- Second AP: 4, 6
Alternatively, we can also pick other values for \( d_1 \) or \( d_2 \) to find other possibilities, but the most straightforward APs between 2 and 8 with equal steps are 4 and 6.
In conclusion, one set of two arithmetic progressions that can be inserted between 2 and 8 is: 2, 4, 6, 8.