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Q.

If the sum of the integers between 100 and 200 that are divisible by 9 is 1683. Then the  sum of the integers between 100 and 200 that are not divisible by 9 is[[1]].


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Detailed Solution

If the sum of the integers between 100 and 200 that are divisible by 9 is 1683. Then the  sum of the integers between 100 and 200 that are not divisible by 9 is 13167.
Given, the sum of the integers between 100 and 200 that are divisible by 9 is 1683.
Thus, the integers are 101,102,103.....199 .
Using 101= a, 1 = d and 199 = a n   in the equation,
  a n = a+ n−1 d  .
199=101+ n−1 1 n−1 1=98 n=99  .
The sum of the integers not divisible by 9 are,
S n = n 2 2a+ n−1 d  .
By putting the values of a, d and n,
S 99 = 99 2 2 101 + 99−1 1 S 99 = 99 2 202+98 S 99 = 99 2 300 S 99 =14850  .
So, the sum of integers between 100-200 not divisible by 9 will be,
S= S 99 − S 11 S=14850−1683 S=13167  .
Therefore, the sum of integers between 100-200 that is not divisible by 9 is 13167.
 
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