A sequence of three numbers a, b, c form an arithmetic sequence if the difference between successive terms in the sequence is the same. That is, when b-a=c-b.
(a) The sequence 2, b, 8 forms an arithmetic sequence. Determine b.
(b) Given a sequence a, b, c, let d_{1} be the non-negative number to increase or decrease b by so that, without changing a or c, the result is an arithmetic sequence. Let d_{2} be the positive number to increase or decrease c by so that, without changing a or b, the result is an arithmetic sequence.
For example, if the three-term sequence is 3,10,13, then we need to decrease 10 to 8 to make the arithmetic sequence 3,8,13. We decreased b by 2, so d_{1}=2. If we change the third term, we need to increase 13 to 17 to make the arithmetic sequence 3,10,17. We increased 13 by 4, so d_{2}=4.
Suppose the original three term sequence is 1,13,17. Determine d_{1} and d_{2}.
(c) Define d_{1}, d_{2} as in part (b). For all three-term sequences, prove that 2 d_{1}=d_{2}.