Understanding Arithmetic Progression Properties

An Arithmetic Progression (AP) is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference.

If three numbers \(a, b, c\) are in AP, it means that the difference between the second and first term is equal to the difference between the third and second term. Mathematically, this is expressed as:

Rearranging this equation, we get:

This condition, \(2b = a + c\), is the fundamental property we will use to check if a given sequence of three terms is in AP.

The question asks us to verify three statements about sequences formed by transforming the terms \(a, b, c\), where \(a, b, c\) are already in AP and \(k\) is a non-zero real number.

Analysis of Statement 1: Scaling AP Terms

Statement 1 says that if \(a, b, c\) are in AP, then \(ka, kb, kc\) are also in AP for any real number \(k \ne 0\).

To check if \(ka, kb, kc\) are in AP, we need to see if \(2(kb) = ka + kc\).

We know that \(a, b, c\) are in AP, so \(2b = a + c\).

Let's multiply the equation \(2b = a + c\) by \(k\). Since \(k\) is a real number, we can multiply both sides:

This is exactly the condition required for \(ka, kb, kc\) to be in AP. Thus, Statement 1 is correct.

Analysis of Statement 2: Shifting AP Terms

Statement 2 says that if \(a, b, c\) are in AP, then \(k - a, k - b, k - c\) are also in AP for any real number \(k \ne 0\).

To check if \(k - a, k - b, k - c\) are in AP, we need to see if \(2(k - b) = (k - a) + (k - c)\).

Let's expand the right side of the equation:

So, the condition becomes \(2(k - b) = 2k - (a + c)\).

Let's expand the left side:

So, we need to check if \(2k - 2b = 2k - (a + c)\).

We know that \(a, b, c\) are in AP, which means \(2b = a + c\).

Substitute \(2b\) with \((a + c)\) on the left side:

Both sides are equal. Thus, Statement 2 is correct. Adding or subtracting a constant from each term of an AP results in another AP.

Analysis of Statement 3: Dividing AP Terms

Statement 3 says that if \(a, b, c\) are in AP, then \(\frac{a}{k}, \frac{b}{k}, \frac{c}{k}\) are also in AP for any real number \(k \ne 0\).

To check if \(\frac{a}{k}, \frac{b}{k}, \frac{c}{k}\) are in AP, we need to see if \(2\left(\frac{b}{k}\right) = \frac{a}{k} + \frac{c}{k}\).

We know that \(a, b, c\) are in AP, so \(2b = a + c\).

Since \(k \ne 0\), we can divide the equation \(2b = a + c\) by \(k\):

This is exactly the condition required for \(\frac{a}{k}, \frac{b}{k}, \frac{c}{k}\) to be in AP. Thus, Statement 3 is correct.

Conclusion on AP Properties

Based on our analysis, all three statements are correct:

• If \(a, b, c\) are in AP, then \(ka, kb, kc\) are in AP (scaling by a non-zero constant).
• If \(a, b, c\) are in AP, then \(k - a, k - b, k - c\) are in AP (subtracting from a constant, which is equivalent to multiplying by -1 and adding k).
• If \(a, b, c\) are in AP, then \(\frac{a}{k}, \frac{b}{k}, \frac{c}{k}\) are in AP (dividing by a non-zero constant).

Since statements 1, 2, and 3 are all correct, the correct option is the one that includes 1, 2, and 3.

Revision Table: AP Properties

Additional Information on Arithmetic Progressions

An arithmetic progression (AP) is defined by its first term (often denoted as \(a_1\) or \(a\)) and its common difference (\(d\)). The terms in an AP follow a linear pattern.

• Common Difference (d): The difference between any term and its preceding term. For the sequence \(a_1, a_2, a_3, \dots\), \(d = a_2 - a_1 = a_3 - a_2 = \dots\).
• General Term (n-th term): The formula for the \(n\)-th term of an AP is \(a_n = a_1 + (n-1)d\).
• Sum of First n Terms (\(S_n\)): The sum of the first \(n\) terms of an AP can be calculated using the formulas:
  • \(S_n = \frac{n}{2}(a_1 + a_n)\) (where \(a_n\) is the last term)
  • \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\)
• Arithmetic Mean: If \(a, b, c\) are in AP, then \(b\) is the arithmetic mean of \(a\) and \(c\), i.e., \(b = \frac{a+c}{2}\). This is equivalent to the condition \(2b = a+c\) used in the problem.

The properties explored in this problem (scaling and shifting) show that APs behave predictably under these basic arithmetic operations. If you perform the same operation (add, subtract, multiply by non-zero, divide by non-zero) on every term of an AP, the resulting sequence remains an AP.