All the page numbers in a book are added, beginning from page 1. However, a page number was added twice by mistake. If the sum obtained was 850, which page number must have been added twice?a)25b)30c)35d)40Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Let the total number of pages in the book be ‘n’.
Let page number ‘x’ be repeated.

⇒ x + n(n+1) /2 = 850
So, n(n+1)/2 ≤ 850
⇒ n² + n ≤ 1700
The greatest possible value of n = 40.
Now, for n = 40, n(n+1)/2 = 40×41/2 = 820
So, x = 30.

Answer

To solve this problem, we need to find the page number that was added twice by using the given information that the sum of all the page numbers is 850.

Let's assume that there are n pages in the book. We know that the sum of all the page numbers from 1 to n is given by the formula for the sum of an arithmetic series:

Sum = (n/2) * (first term + last term)

In this case, the first term is 1 and the last term is n. So we can rewrite the formula as:

850 = (n/2) * (1 + n)

Simplifying this equation, we get:

850 = (n^2 + n) / 2

Multiplying both sides of the equation by 2, we obtain:

1700 = n^2 + n

Rearranging the equation, we get a quadratic equation:

n^2 + n - 1700 = 0

To solve this equation, we can factorize it or use the quadratic formula. In this case, let's use the quadratic formula:

n = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 1, b = 1, and c = -1700. Plugging in these values, we get:

n = (-1 ± √(1^2 - 4(1)(-1700))) / (2(1))

Simplifying further, we have:

n = (-1 ± √(1 + 6800)) / 2

n = (-1 ± √(6801)) / 2

Since we are looking for a positive integer value for n, we can disregard the negative solution.

Taking the positive square root, we get:

n = (-1 + 82) / 2 = 81 / 2 = 40.5

Since the number of pages in a book is always a whole number, we conclude that n = 40.

Therefore, the page number that was added twice is 40, which corresponds to option B.

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