How to Find the First Negative Term of an AP

What is the method?

Write the nth term as a + (n − 1)d and solve a + (n − 1)d < 0. Choose the smallest positive integer n satisfying this inequality. Then calculate both that term and the preceding term to prove you found the first negative value.

Example: 25, 21, 17, …

The first term is 25 and the common difference is −4. We need 25 − 4(n − 1) < 0. Expanding gives 29 − 4n < 0, so n > 29/4 = 7.25. The smallest integer greater than 7.25 is 8.

The eighth term is 25 − 28 = −3. The seventh term is 25 − 24 = 1. This adjacent-term check establishes that the eighth term is the first negative term, rather than merely a negative term.

Why zero changes the answer

For 24, 20, 16, …, solve 28 − 4n < 0, giving n > 7. The seventh term equals zero, so it does not qualify. The eighth term, −4, is the first negative term. Replacing < with ≤ would answer a different question: the first non-positive term.

Check the direction of the progression

If a is already negative, the first negative term is term 1. If a is non-negative and d is zero or positive, there is no negative term. For a decreasing AP with a non-negative starting term, the strict-inequality approach applies. If the progression has a specified final term, also check that the required position lies inside it.

Practice

1. Find the first negative term of 10, 7, 4, … . Term 5 is −2; term 4 is 1.
2. Find it for 12, 9, 6, … . Term 6 is −3; term 5 is zero.
3. Does 0, 2, 4, … have a negative term? No.

When dividing an inequality by a negative number, reverse its sign. A final answer should include both the position and the term value.

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