# 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

1
by chandanasnehal

2015-05-06T03:37:03+05:30

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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

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