Answer:U₁ = 2
.
Explanation:
Un+1 = 2U₁ + 6 (Equation 1)
Uo = 10
We want to find U₁. We can start by using Equation 1 with n = 1 to get:
U2 = 2U₁ + 6
We can then use the value of Uo = 10 to find U₂ as follows:
U₂ = 2U₁ + 6
U₂ = 2U₁ + 2(3)
U₂ = 2(U₁ + 3)
We can then use the value of U₂ to find U₃:
U₃ = 2U₂ + 6
U₃ = 2(2U₁ + 2(3)) + 6
U₃ = 4U₁ + 12 + 6
U₃ = 4U₁ + 18
We can keep using this process to find Un in terms of U₁, until we reach the value of n we need. For example, we can find U₄ as follows:
U₄ = 2U₃ + 6
U₄ = 2(4U₁ + 18) + 6
U₄ = 8U₁ + 36
So we have:
Uo = 10
U₂ = 2(U₁ + 3)
U₃ = 4U₁ + 18
U₄ = 8U₁ + 36
and so on.
To find U₁, we need to use the equation for U₂:
U₂ = 2(U₁ + 3)
Substituting U₂ = 10 (from the given value of Uo), we get:
10 = 2(U₁ + 3)
Simplifying, we get:
5 = U₁ + 3
Subtracting 3 from both sides, we get:
U₁ = 2
Therefore, U₁ = 2.