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Let 3,7-b and -3 a^2+10 a+2 be three natural numbers, which are first three terms (in order) of an A.P.. If (a, b) I , then find the number of possible such arithmetic progressions.

Correct answer

4

Step-by-step solution

aligned & 3,7-b, -3 a^2+10 a+2 A.P. & -3 (a^2-4+4 )+10 a+2 =-3(a-2)- 2 a+2 & a+2= 1, 2, a=-1,-3,0,-4 & a=-1 3,7-b, 7 b=2 & a=-3 3,7-b, 17 b=-3 & a=0 3,7-b, 5 b=3 & a=-4 3,7-b, 19 b=-4 aligned Number of A.P.'s are 4 .

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