Prove that the sum of any number of terms of the arithmetic sequence 16, 24, 32, … starting from the first, added to 9 gives a perfect square.

a = 16

d = 24-16 = 8

Let number of terms = n

Sn =

=

= n(12 + 4n)

Sn + 9 = 4n2 + 12n + 9 = (2n + 3)2

Hence, the sum of any number of terms of the arithmetic sequence 16, 24, 32, …. starting from the first, added to 9 gives a perfect square.

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