9514 1404 393
Answer:
squares are of the form 3n or 3n+1; the sequence is of the form 3n-1, so none of the sequence will be a square
Explanation:
The given arithmetic sequence has first term 2 and common difference 3, so its explicit formula is ...
an = 2 +3(n -1) = 3n -1 . . . . for counting numbers n
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All integers are of one of these forms: 3n-1, 3n, 3n+1, for some integer n. The squares of these are ...
(3n -1)² = 9n² -6n +1 = 3(3n² -2) +1 = 3k+1 for some k
(3n)² = 3(3n²) = 3k for some k
(3n +1)² = 9n² +6n +1 = 3(3n² +2) +1 = 3k+1 for some k
Note that none of these squares is of the form 3n -1.
Hence, the square of an integer cannot be in the given sequence.