Finding the First Negative Term in an Arithmetic Progression

The given sequence is an Arithmetic Progression (A.P.). An A.P. is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference (d).

Identifying the A.P. Parameters

Let's identify the first term and the common difference of the given A.P.:

• The first term, \(a_1\) or \(a\), is the first number in the sequence. Here, \(a = 129\).
• The common difference, \(d\), is found by subtracting any term from its succeeding term.

Calculating the common difference:

• \(d = 120 - 129 = -9\)
• \(d = 111 - 120 = -9\)
• \(d = 102 - 111 = -9\)

The common difference is \(d = -9\). Since the common difference is negative, the terms of the A.P. are decreasing.

Using the nth Term Formula

The formula for the \(n\)th term of an Arithmetic Progression is given by:

\( a_n = a + (n-1)d \)

We are looking for the first negative term. A term is negative when its value is less than 0. So, we need to find the smallest integer value of \(n\) for which \(a_n < 0\).

Setting up and Solving the Inequality for Negative Terms

We want to find \(n\) such that:

\( a_n < 0 \)

Substitute the values of \(a\) and \(d\) into the formula:

\( 129 + (n-1)(-9) < 0 \)

Now, let's solve this inequality step-by-step:

• Distribute the \(-9\):
  \( 129 - 9(n-1) < 0 \)
  \( 129 - 9n + 9 < 0 \)
• Combine the constant terms:
  \( (129 + 9) - 9n < 0 \)
  \( 138 - 9n < 0 \)
• Add \(9n\) to both sides of the inequality:
  \( 138 < 9n \)
• Divide both sides by 9:
  \( \frac{138}{9} < n \)

Simplify the fraction \(\frac{138}{9}\). Both 138 and 9 are divisible by 3:

\( \frac{138 \div 3}{9 \div 3} = \frac{46}{3} \)

So, the inequality is:

\( \frac{46}{3} < n \)

Convert the fraction to a decimal:

\( 15.333... < n \)

Determining the First Integer Term Number

The inequality \(15.333... < n\) tells us that the term number \(n\) must be greater than 15.333.... Since \(n\) must be a whole number (you can't have a fraction of a term), the smallest integer value of \(n\) that satisfies this condition is 16.

This means the 16th term is the first term in the sequence that will have a value less than zero (i.e., be negative).

Conclusion

The first negative term of the given A.P. (129, 120, 111, 102, ...) is the 16th term.

Revision Table: Arithmetic Progression Concepts

Additional Information on Finding Terms in A.P.

When working with Arithmetic Progressions, understanding the common difference is key. If the common difference is positive, the A.P. is increasing. If it's negative, the A.P. is decreasing. Our example had a negative common difference (\(-9\)), so the terms were getting smaller, eventually becoming negative.

To find the first negative term in a decreasing A.P., we essentially find where the terms cross from positive to negative. The inequality \(a_n < 0\) helps us pinpoint this crossover point in terms of the term number \(n\). Since the term number must be an integer, we always round up to the next whole number if the solution to the inequality is not an integer itself. This gives us the position of the very first term that is negative.