Question

Find all arithmetic progressions of natural numbers beginning with 3 whose sum is a three digit number whose digits form a non constant geometric progression.

Answer 1

Ans: Consider the arithmetic progression 3, a2, ...,an _ d and the sum Such that common difference= a ак-1 n [6(n 1d] 2 [2а, + (n - 1)d] 2 Sn Since the sum is a three digit number which form a non constant geometric progression so the answer is either 124, 139, 248, 421, 842 or 931. Thus to solve n [(n 1)d M 2 Зп + or, бn + n?d — nd 2M 931 where M E 124,139,248,421,842 or M can be described as 100b 10rb + r2b 7 We find one such solution 27 43 3 11 19 35 51 59 whose sum is 248