Institution: AGRANI BANK LTD Exam: Agrani bank Ltd - Senior Officer - 12.08.2011
Topic: Mathematics (সাধারণ গণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

To solve the problem, we need to consider that the lengths of the three sides of a triangle need to be consecutive integers, say \(n\), \(n+1\), and \(n+2\), where \(n\) is a positive integer. We need to satisfy two main conditions: the lengths must be positive integers, and they must obey the triangle inequality rules.

The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. We can express this principle mathematically:

1. \(n + (n + 1) > (n + 2)\) simplifies to \(2n + 1 > n + 2\), which reduces to \(n > 1\).
2. \(n + (n + 2) > (n + 1)\) simplifies to \(2n + 2 > n + 1\), leading to \(n > -1\), which is always true for positive \(n\).
3. \((n + 1) + (n + 2) > n\) simplifies to \(2n + 3 > n\), reducing to \(n > -3\), again always true for positive integers.

Starting from the smallest integer \(n=1\): we have side lengths as \(1\), \(2\), and \(3\). The sum is \(1 + 2 + 3 = 6\), which doesn't satisfy the triangle inequality. For \(n=2\), the side lengths are \(2\), \(3\), and \(4\), giving a sum of \(2 + 3 + 4 = 9\) – still invalid.

However, when \(n=3\), the sides are \(3\), \(4\), and \(5\) with a sum of \(3 + 4 + 5 = 12\). Finally, at \(n=4\), the sides become \(4\), \(5\), and \(6\), summing to \(4 + 5 + 6 = 15\), which satisfies the triangle inequalities.

Thus, the smallest possible value of the sum of the lengths of the three sides that form a valid triangle is indeed **15** (option D).