Let a 1 , a 2 , a 3 , .....be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b 1 , b 2 , b 3 ,.....be a sequence of positive integers in geometric progression with common ratio 2. If a 1 = b 1 = c, then the number of all possible values of c, for which the equality 2(a 1 + a 2 +.....+ a n ) = b 1 + b 2 +.....+ b n holds for some positive integer n, is_____
Text Solution
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(1)
2(a 1 + a 2 +.....+ a n ) = b 1 + b 2 +.....+ b n
2 x-2 >x 2 -x if x 
8
2 x -1>2x if x>3
So, 
as c 
1 values of c are not possible for n 
8
possible values of n = 3, 4, 5, 6, 7; only satisfy when n = 3 and c = 12
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