Arithmetic Progressions Basic Understanding and Formation

Theory: Patterns such as salary increments, decreasing lengths of ladder rungs, or yearly savings can all form arithmetic progressions when each term increases or decreases by a fixed amount.

Example: Question: Shakila saves ₹100 on her daughter’s first birthday and increases the amount by ₹50 every year. List the amounts saved for the first five years.

Solution:  The sequence is: 100, 150, 200, 250, 300
This is an AP with first term \(a = 100\) and common difference \(d = 50\).

Theory: In an arithmetic progression (AP), the difference between any two consecutive terms remains constant and is called the common difference, denoted by d.
Mathematically, \(d = a_{k+1} – a_k\)

Example:

Question: Find the common difference of the sequence: 3, 5, 7, 9, …

Solution:  d = 5 – 3 = 2. Since the difference is constant, it is an AP with d = 2.

Theory: An arithmetic progression is a sequence of numbers where each term (after the first) is obtained by adding a constant (the common difference) to the previous term.

Example:

Question: Is the sequence 4, 7, 10, 13 an AP?

Solution:  Yes 7 – 4 = 3, 10 – 7 = 3, 13 – 10 = 3. Constant difference = 3.

Theory: A finite AP has a specific number of terms and ends with a last term.
An infinite AP continues indefinitely without a last term.

Example:

Question: Is  5, 10, 15, 20 a finite or infinite AP?

Solution:  It’s finite (4 terms). If it continued beyond, it would be infinite.

Theory: Each number in an AP is a term. The first term is a, second is a + d, third is a + 2d, and so on.

Example:

Question: What is the 4th term of an AP where a = 2, d = 3?

Solution:  \(a_4 = a + 3d = 2 + 3×3 = 11\)

Theory: The general form of an AP is:
\(a, a + d, a + 2d, a + 3d, …, a + (n-1)d\)

Example:

Question: Write the first 4 terms of an AP where a = 5, d = -2

Solution:  5, 3, 1, -1

Theory: Given any sequence, if the difference between consecutive terms is constant, it forms an AP. The first term is the initial number, and d is the difference.

Example:

Question: Is 2, 4, 6, 9 an AP?

Solution:  4 – 2 = 2, 6 – 4 = 2, 9 – 6 = 3 → Not an AP (d is not constant)

Theory: Physical arrangements like increasing plant rows or tile lengths can be modeled using AP where quantity changes uniformly.

Example:

Question: A ladder has rungs decreasing in length by 2 cm from bottom to top. The bottom rung is 45 cm. What is the length of the 4th rung?

Solution:  AP: 45, 43, 41, 39 → 4th term is 39 cm

Theory: Sequences involving whole numbers, integers, fractions, decimals can all be APs if they follow constant difference.

Example:

Question: Is the sequence 0.5, 0.0, -0.5, -1.0 an AP?

Solution:  Yes. Each term decreases by 0.5. Common difference d = -0.5

Theory: If the difference between consecutive terms is not constant, the sequence is not an AP.

Example:

Question: Is 1, 2, 4, 7, 11 an AP?

Solution:  2-1 = 1, 4-2 = 2, 7-4 = 3 → Differences vary → Not an AP

Theory: Arithmetic Progressions model situations involving linear growth or decay such as salary increments, savings, penalties, and installment schemes.

Example:

Question: A loan reduces by ₹1000 every month. If the initial amount is ₹10,000, what is the balance after 3 months?

Solution:  AP: 10000, 9000, 8000, 7000 → Balance after 3 months = ₹7000