Arithmetic Progression (A.P.)

An Arithmetic Progression (A.P.) is a list of numbers in which each term is obtained by adding the same fixed number to the previous term.

Real-life analogy:
If you save ₹500 in the first month, ₹700 in the second month, ₹900 in the third month, and so on, your monthly savings increase by ₹200 every time. This forms an A.P.

Here, the fixed increase is ₹200.

Important: The fixed number added is called the common difference.

An Arithmetic Progression (A.P.) is a sequence in which the difference between consecutive terms is constant.

• Common difference = d = second term − first term

• The general form of an AP is a, a + d, a + 2d, a + 3d, …
  a = first term
  d = common difference

Check whether the given sequence is an A.P. If it is an A.P., find the next two terms.

Solution:

Step 1: Identify the terms of the sequence

Given sequence: \[ \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2}, \ldots \]

\[ t_{1} = \frac{3}{2}, t_{2} = \frac{1}{2}, t_{3} = -\frac{1}{2}, t_{4} = -\frac{3}{2} \]

Step 2: Find the common differences ( \[ t_{n + 1} - t_{n} \] )

\[ t_{2} - t_{1} = \frac{1}{2} - \frac{3}{2} = -\frac{2}{2} = -1 \]

\[ t_{3} - t_{2} = -\frac{1}{2} - \frac{1}{2} = -\frac{2}{2} = -1 \]

\[ t_{4} - t_{3} = -\frac{3}{2} - \left(-\frac{1}{2}\right) = -\frac{3}{2} + \frac{1}{2} = -\frac{2}{2} = -1 \]

Step 3: Confirm the A.P.

Since the common difference \[ d = -1 \] is constant, the given sequence is an A.P.

Step 4: Find the next two terms

\[ -\frac{3}{2} - 1 = -\frac{5}{2} \]

\[ -\frac{5}{2} - 1 = -\frac{7}{2} \]

\[ \therefore \] The next two terms are \[ -\frac{5}{2} \] and \[ -\frac{7}{2}. \]

• Definition: An A.P. is a sequence in which the difference between consecutive terms is constant.
• Condition for A.P.: \[ a_{n + 1} - a_{n} = d \]
• General form: \[ a,\; a + d,\; a + 2d,\; a + 3d, \ldots \]