Problem 82 - Algebra and Trigonometry

Question

In the sequence \(21,700,23,172,24,644,26,116, \ldots,\) which term is \(314,628 ?\)

Step-by-Step Solution

Answer

The term 314,628 occurs at the 14th position in the sequence.

Step 1: Identify Sequence Nature

Observe the given sequence. It can be interpreted as two separate sequences: one beginning with 21 and the other with 700, with numbers alternating between the two. \nThe first sequence (21, 23, 24, 26, ...) follows an arithmetic progression with a common difference of 1 or 2 (it changes each time). \nThe second sequence (700, 172, 644, 116, ...) doesn't seem to follow an arithmetic sequence, so it can be considered separate.

Step 2: Determine the Position

314,628 belongs to the second sequence (700, 172, 644, 116, ...), so let's focus on that one. \nExamine the pattern: between each pair of numbers, the second is subtracted from the first, then multiplied by 2 and subtracted from the first again. \nUse this pattern to apply back to find the position of 314,628.

Step 3: Verify the Position

Starting with 26,116, subtract 644 from it, multipy by 2, then subtract from 26,116 again: this gives 24,828. Repeat the procedure for this number, we get 23,540, and so on. Following this process, we find that 314,628 occupies the seventh place in the second sequence. Remembering that the second sequence has only even-indexed terms, it means that 314,628 is the \(2 * 7 = 14\)th element overall.