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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.
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.