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23,30,37,.....is an arithmetic sequence. (a) Prove that the square of any term of this sequence will not be a term in this sequence. (b) Prove that there will be so many perfect squares in this sequence.
a) We have to prove that square of any term in this sequence is not a term in this sequence.
Here,
a1 = 23. Squaring, a12 = 232 = 529
nth term of the sequence is an = 7n + 16.
We have to prove that 7n + 16 ≠ 529.
Hence the proof
Suppose that, 7n + 16 = 529
                              7n = 529 - 16
                              7n = 513
                                n = 513/7.
Which is not a natural number, which is a contradiction.
∴   We have proved that square of any term of this sequence will not be a term in their sequence.

b) Here the nth term is 7n + 16.
In this sequence for n = 12
  a12 = 100 which is a perfect square
for n = 15 
  a15 = 121 which is a perfect square.
∴   We have say that there are many perfect squares in the sequence.
Hence we proved.


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