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In a sequence of positive integers each term after the first is 1/3 of the sum of the term that precedes it and the term that follows it in the sequence. What is the 5th term of this sequence if the 1st term is 2 and the 4th term is 34?

Nov 5, 2019

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the first is 1/3 of the sum of the term that precedes it and the term that follows it in the sequence

2, x, y, 34

That is the sequence we have so far

Lets model an equation after that:

$$\frac{1}{3}(2+y)=x$$

$$\frac{1}{3}(x+34)=y$$

solve(1/3(2+y)=x, (1\3)(x+34)=y = {y=13, x=5}

Solving, we get the 3rd term being equal to 13.

Knowing that the third term is 13, the fourth term is 34, we can calculate the 5th term as follows:

$$\frac{1}{3}(13+x)=34$$

With x being the 5th term

(13+x)/3=34 = x=89

$$\boxed{89}$$

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Nov 5, 2019