Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 4 of 8
Arithmetic Progressions
“Arithmetic progressions grow by equal steps because consistency is their personality.”
• Identify arithmetic progressions. • Find the common difference. • Derive and use the nth-term formula. • Write recursive rules for APs. • Connect APs with straight-line graphs and daily-life situations.
Introductions
Some number sequences follow a simple and regular pattern: the same number is added each time to get the next term. For example, in the sequence 1, 5, 9, 13, 17, … each term is 4 more than the previous one. The difference between consecutive terms remains the same throughout the sequence.
A sequence with this kind of constant change is called an arithmetic progression. The fixed number that is repeatedly added or subtracted is called the common difference. Recognising this constant difference helps us understand the pattern, predict later terms, and describe the sequence using a general rule.
A sequence in which the difference between consecutive terms is constant.
The fixed number added to one term to obtain the next term in an arithmetic progression.
Problem
Compare 1,4,7,10,… and 11,7,3,−1,… .
- 1.First sequence: differences are +3, so d=3.
- 2.Second sequence: differences are −4, so d=−4.
- 3.Both are APs because the difference stays constant.
Deriving the nth Term
Start with first term a. The second term is a+d. The third is a+2d. The fourth is a+3d. So the nth term has added d exactly n−1 times.
Problem
Find the 20th term of 3,8,13,18,… .
- 1.a=3 and d=5.
- 2.t₂₀=3+(20−1)5.
- 3.t₂₀=3+95=98.
Recursive Form of an AP
Visualising an AP
If the position n is plotted on the x-axis and the term value on the y-axis, points from an AP lie on a straight line. This happens because tₙ=a+(n−1)d is a linear rule in n.
Daily-Life Example: Taxi Fare
Suppose a taxi has a fixed booking fee and then adds the same charge for every kilometre. The total fares for 1 km, 2 km, 3 km, … form an AP because each extra kilometre adds the same amount.
Problem
A taxi charges a fixed ₹180 plus ₹35 per kilometre. Find the fare for 1 km, 2 km and 10 km.
- 1.1 km: 180+35=₹215.
- 2.2 km: 180+70=₹250.
- 3.The sequence begins 215,250,285,… with d=35.
- 4.For n km, fare=180+35n.
- 5.For 10 km, fare=₹530.
Practice Problems
- Check whether 4,9,14,19,… is an AP and find d.
- Find the nth term of 12,8,4,0,… .
- Find the 30th term of 7,10,13,16,… .
- Which term of 5,11,17,23,… is 173?
- Write both explicit and recursive rules for −3,2,7,12,… .
- A salary starts at ₹4,80,000 and increases by ₹25,000 each year. Write the sequence for the first five years and identify a and d.
Key Takeaways
• APs have a constant difference. • The common difference may be positive, zero or negative. • tₙ=a+(n−1)d. • APs can also be written recursively. • When position is plotted against term value, AP points lie on a straight line.
Next, we derive a beautiful shortcut for 1+2+3+…+n and connect it to triangular numbers.