Today we will explore a special type of sequence called a jump sequence. A sequence is defined as a jump sequence if for any three consecutive terms, the sum of the first and third terms equals twice the middle term. This condition creates a specific arithmetic relationship that we will investigate through various examples and proofs.
Now let's test two specific sequences to see if they are jump sequences. First, we examine the arithmetic sequence a_n equals 2n minus 1. The first few terms are 1, 3, 5, 7, 9. Let's check: a_1 plus a_3 equals 1 plus 5 equals 6, and 2 times a_2 equals 2 times 3 equals 6. Since they are equal, this arithmetic sequence satisfies the jump sequence condition. Next, we test the geometric sequence a_n equals 2 to the power n. The terms are 2, 4, 8, 16, 32. Checking: a_1 plus a_3 equals 2 plus 8 equals 10, but 2 times a_2 equals 2 times 4 equals 8. Since 10 does not equal 8, this geometric sequence is not a jump sequence.
Now we tackle part 2, which involves proving a necessary and sufficient condition. Given a sequence b_n where the difference between consecutive terms follows the pattern b_{n+1} minus b_n equals negative one to the power n, we need to prove that this sequence is a jump sequence if and only if b_1 equals one half. First, we find the general term by summing the differences. The sum of alternating ones gives us a piecewise function: b_n equals b_1 when n is odd, and b_1 minus 1 when n is even. Next, we apply the jump sequence condition. Substituting our piecewise formula into the condition b_n plus b_{n+2} equals 2 times b_{n+1}, we can solve for b_1 and find that it must equal one half for the sequence to be a jump sequence.
Finally, we solve part 3, which asks for the range of the first term c_1 in a jump sequence that is also non-decreasing. We have two conditions: the jump sequence condition c_n plus c_{n+2} equals 2 times c_{n+1}, and the non-decreasing condition c_{n+1} greater than or equal to c_n. From the jump sequence condition, we can express c_{n+2} as 2c_{n+1} minus c_n. Rearranging this, we find that c_{n+2} minus c_{n+1} equals c_{n+1} minus c_n, which means the differences between consecutive terms form an arithmetic sequence. For the sequence to be non-decreasing, all differences must be non-negative. Through careful analysis of these constraints, we determine that the first term c_1 must be greater than or equal to zero.
To conclude our exploration of jump sequences, let's summarize the key findings. In part 1, we discovered that arithmetic sequences are jump sequences, but geometric sequences are not. Part 2 revealed that for sequences with alternating differences, being a jump sequence is equivalent to having a first term of one half. Part 3 showed that non-decreasing jump sequences require the first term to be non-negative. The fundamental insight is that jump sequences impose a linear constraint between any three consecutive terms, creating a harmonic relationship that significantly restricts the possible behavior of such sequences. This condition, a_n plus a_{n+2} equals 2 times a_{n+1}, is both simple in form and profound in its implications for sequence theory.