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Find the sum of all natural numbers between 100 to 200 and which are not divisible by 4 .

Solution:
Your problem `->` Find the sum of all natural numbers between 100 to 200 and which are not divisible by 4 .


Required Addition = `(100 + 101 + 102... + 200) - (100 + 104 + 108... + 200)`

Required Addition = `S' - S''`

We know that, `S_n = n/2 (a + l)`

`:. S_101 = 101/2 * (100 + 200)`

`= 101/2 (300)`

`= 15150`

Numbers between `100` and `200` divisible by `4` are `100, 104, 108 ...`

Which are in arithmetic progression.
In which `a=100` and `d=4`

Let `n` be the term such that `f(n) = 200`

We know that, `f(n) = a + (n - 1)d`

`100 + (n - 1)(4) = 200`

`(n - 1)(4) = 100`

`n - 1 = 25`

`n = 26`

We know that, `S_n = n/2 [2a + (n - 1)d]`

`:. S_26 = 26/2 * [2(100) + (26 - 1)(4)]`

`= 13 * [200 + (100)]`

`= 13 * [300]`

`= 3900`

`:.` Required Addition = `15150 - 3900`

`= 11250`






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