Solution:

Let's assume that the torn page is "x".

The sum of the numbers on the remaining pages is 195.

We can use the formula to find the sum of consecutive integers from 1 to n:

Sum = n(n+1)/2

We need to find the sum of the numbers on the remaining pages, which is 195. So, we can write:

195 = n(n+1)/2

Solving this equation, we get n = 19.

Therefore, the booklet had 19 pages.

Now, we need to find the torn page. We can use the fact that the sum of consecutive even (or odd) integers is twice the sum of consecutive integers. So, we can find the sum of consecutive even (or odd) integers from 1 to 19, and then subtract the sum of the remaining pages (which is 195) to get the torn page.

Sum of consecutive even integers from 1 to 19:

2+4+6+...+18 = 2(1+2+3+...+9) = 2(9x10/2) = 90

Subtracting the sum of the remaining pages (195) from the sum of consecutive even integers from 1 to 19 (90), we get:

90 - 195 = -105

This means that there is an odd number difference between the sum of consecutive even integers from 1 to 19 and the sum of the remaining pages. This difference is equal to the torn page number.

Now, we need to find an odd number that is less than or equal to 19 (the number of pages in the booklet) and has a difference of 105 with the sum of consecutive even integers from 1 to 19.

We can start by trying odd numbers that are less than or equal to 19 and see which one satisfies the condition. We can write the sum of consecutive odd integers from 1 to n as:

Sum = n^2

Trying n = 1, we get Sum = 1, which is not equal to 105. So, we can try n = 3.

Sum of consecutive odd integers from 1 to 3:

1+3 = 4

Square of 3:

3^2 = 9

Difference between the sum of consecutive odd integers from 1 to 3 and the square of 3:

9 - 4 = 5

This difference is not equal to 105. So, we can try n = 5.

Sum of consecutive odd integers from 1 to 5:

1+3+5 = 9

Square of 5:

5^2 = 25

Difference between the sum of consecutive odd integers from 1 to 5 and the square of 5:

25 - 9 = 16

This difference is not equal to 105. So, we can try n = 7.

Sum of consecutive odd integers from 1 to 7:

1+3+5+7 = 16

Square of 7:

7^2 = 49

Difference between the sum of consecutive odd integers from 1 to 7 and the square of 7:

49 - 16 = 33

This difference is not equal to 105. So, we can try n = 9.

Sum of consecutive odd integers from 1 to 9:

1+3+5+7

Sum of N natural numbers = n(n+1)/2

Here after removing pages, sum of numbers on remaining pages = 195
so, the number before removing pages would definitely be greater than 195
so , taking n= 20 , = 20(20+1)/2 = 420/2= 210 (closest & greater to 195)

so, the student forget to add = 210-195 = 15

So , the two subsequent numbers whose addition is 15 = 7, 8