If the 11th and the 8th terms of an arithmetic progression are in the ratio 2 : 5, which of the following statements is definitely false?The sum of no two consecutive terms is zero.The sum of no three consecutive terms is zero.At least one of the terms is zero.None of the above

In an arithmetic progression (AP), the difference between any two successive terms is constant. This constant difference is called the common difference.

Given that the 11th and 8th terms of an AP are in the ratio 2:5, we can write this as:

2/5 = (a + 10d) / (a + 7d)

where a is the first term and d is the common difference.

Solving this equation, we get:

2a + 20d = 5a + 35d

Simplifying, we get:

3a = 15d

So, a = 5d

Now, let's examine the statements:

1. The sum of no two consecutive terms is zero.

This statement can be false. If the common difference is not zero, there can be two consecutive terms whose sum is zero. For example, if a = 5 and d = -5, then the first term is 5 and the second term is 0, so their sum is zero.

2. The sum of no three consecutive terms is zero.

This statement can also be false. If the common difference is not zero, there can be three consecutive terms whose sum is zero. For example, if a = 5, d = -5, then the first three terms are 5, 0, and -5, so their sum is zero.

3. At least one of the terms is zero.

This statement can be true or false, depending on the values of a and d. If a = 5 and d = -5, then one of the terms is zero. But if a = 5 and d = 1, then none of the terms is zero.

4. None of the above

This statement is definitely false, because we have shown that the first two statements can be false and the third statement can be either true or false. Therefore, it is not true that none of the above statements is definitely false.