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All the terms of an AP are natural numbers. the sum of its first nine terms lies between 200 and 220. if second term is 12 then find the common difference

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Let the series be a , a+d, a+2d , .....

sum of the first 9 terms: [2 a + 8 d] * 9/2 = 9 a + 36 d

so 200 <= 9 a + 36 d <= 220

2nd term : a + d = 12 => 9a = 9 (12 - d) = 108 - 9 d

=> 200 <= 108 - 9d + 36 d <= 220

=> 92 <= 25 d <= 112

=> 3.6 <= d <= 4.4

=> d = 4 as d is a natural number and a is also a natural number.

=> a = 12 - 4 = 8.

series: 8, 12, 16, 20, 24, ... sum of the first 9 numbers: 216

sum of the first 9 terms: [2 a + 8 d] * 9/2 = 9 a + 36 d

so 200 <= 9 a + 36 d <= 220

2nd term : a + d = 12 => 9a = 9 (12 - d) = 108 - 9 d

=> 200 <= 108 - 9d + 36 d <= 220

=> 92 <= 25 d <= 112

=> 3.6 <= d <= 4.4

=> d = 4 as d is a natural number and a is also a natural number.

=> a = 12 - 4 = 8.

series: 8, 12, 16, 20, 24, ... sum of the first 9 numbers: 216