jmc

Question

In a sequence of positive integers each term after the first is 13 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?

Accepted Answer

Let a,b,c be 2nd, 3rd, and 5th terms respectively. Our sequence is therefore 2,a,b,34,c,. From the information given, we have align* a &= 13(2+b) b &= 13(a+34) 34 &= 13(b+c). align* Before we find c, we use the first two equations to solve for b. Substituting a = 13(2+b), we obtain align* b &= 13(13(2+b)+34) 3b &= 13(2+b)+34 9b &= 2+b+102 8b &= 104 b &= 13. align* Substituting b = 13 into 34 = 13(b+c), we obtain align* 34 &= 13(13+c) 102 &= 13+c c &= 89. align*