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.

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

To find the value of u4 we replace un by u3 in the equation and then calculate un+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 un+1 is equal to un .

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.

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