SOLUTION: given sequence 2;5;8 2.1.1 if the pattern continues , then write down the next two terms 2.2.2 prove that none of the terms of this sequence are perfect squares

Question 1177105: given sequence 2;5;8
2.1.1 if the pattern continues
, then write down the next two terms
2.2.2 prove that none of the terms of this sequence are perfect squares

Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!
given sequence ;;
2.1.1 if the pattern continues, then write down the next two terms
as you can see, common difference is , so next two terms are and

2.2.2 prove that none of the terms of this sequence are perfect squares
Let each term of the given arithmetic progression be defined as
, where and

Hence, each term of the AP is of the form , ∈ .
We know that any integer is of the form , or for ∈.
Taking , we have , with .
Taking , we have , where
Taking , we have , with
Hence, we can see that a perfect square is always of the form or , where is an integer.
Hence, we can infer that a perfect square is not of the form .
Also, we have each term of the given AP to be of the form .
Hence, of the given AP is a .

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