A sequence is defined by the recurrence relation u(n+1) = 1/3 u(n) + 10 with u(3) = 6 . Find the value of u(4) and the limit of the sequence.

A sequence is defined by the recurrence relation u_n+1 = 1/3 u_n + 10  with u₃ = 6 . Find the value of u₄ and the limit of the sequence.

To find the value of u₄ we replace u_n by u₃ in the equation and then calculate u_n+1

u₄ = 1/3 u₃ + 10

u₄ = 1/3 x 6 + 10

u₄ = 2 + 10

u₄ = 12

To find the limit of the series we have to find for which value u_n+1 is equal to u_n .

Let's call this value x . Then we have:

x = 1/3 x + 10

We can subtract 1/3 x on both sides to get:

x - 1/3 x = 10

2/3 x = 10

Now we multiply by 3 and then divide by 2:

x = 10 x 3 / 2

x = 15

The limit of the sequence is 15.