Sequences — Lesson
1) Hook — The Magic of Saving Coins
Imagine you start saving money by putting ₹10 in your piggy bank on the first day. Every day, you decide to add ₹5 more than the previous day. So, on day 2 you add ₹15, on day 3 ₹20, and so on. How much money will you have saved after 10 days?
This simple idea of adding a fixed amount more each day is a perfect example of a sequence — a list of numbers following a specific pattern. Sequences help us understand patterns in nature, finance, and even technology!
2) Core Concepts — Understanding Sequences
A sequence is an ordered list of numbers, where each number is called a term. The position of a term is given by n (1st term, 2nd term, etc.).
- Arithmetic Sequence (AP): Each term increases or decreases by a constant difference.
- Geometric Sequence (GP): Each term is multiplied or divided by a constant ratio.
- Other sequences: Fibonacci sequence, quadratic sequences, etc.
Arithmetic Progression (AP)
An AP is a sequence where the difference between consecutive terms is constant. This difference is called the common difference (d).
| Term (n) | Sequence (Example: a=3, d=4) |
|---|---|
| 1 | 3 |
| 2 | 7 (3 + 4) |
| 3 | 11 (7 + 4) |
| 4 | 15 (11 + 4) |
Geometric Progression (GP)
A GP is a sequence where each term is obtained by multiplying the previous term by a constant called the common ratio (r).
| Term (n) | Sequence (Example: a=2, r=3) |
|---|---|
| 1 | 2 |
| 2 | 6 (2 × 3) |
| 3 | 18 (6 × 3) |
| 4 | 54 (18 × 3) |
3) Key Formulas / Rules
nth term (Tn): Tn = a + (n - 1)d
Sum of first n terms (Sn): Sn = (n/2) [2a + (n - 1)d] or Sn = (n/2)(a + l), where l is the last term.
nth term (Tn): Tn = a × rn-1
Sum of first n terms (Sn): Sn = a (rn - 1) / (r - 1), for r ≠ 1
4) Did You Know?
Indian mathematician Pingala (circa 3rd century BCE) was one of the earliest to study sequences, especially the Fibonacci sequence, which appears in nature — like the arrangement of leaves, flower petals, and even the spiral shells of snails!
5) Exam Tips — Mastering Sequences for Board Exams
- Identify the type of sequence first: Check if the difference or ratio is constant.
- Write down first few terms: Helps to spot the pattern and avoid mistakes.
- Use correct formula carefully: Remember n - 1 in the term formulas.
- Common mistakes: Mixing up a (first term) and d or r, incorrect sign of d, or forgetting to simplify sums.
- Previous Year Question Pattern:
- Find the nth term of an AP or GP given initial terms.
- Calculate sum of first n terms.
- Word problems involving sequences, e.g., savings, population growth, or interest calculations.
- Derive formulas or prove properties of sequences.
Sequences — Mcq
Sequences — Mnemonic
Mnemonic 1: "AP GP HP - A Great Helper!" 📚✨
Remember the three common sequences with this phrase:
- AP - Arithmetic Progression (difference constant)
- GP - Geometric Progression (ratio constant)
- HP - Harmonic Progression (reciprocals in AP)
Hindi twist: "Arithmetic ka Antar, Geometric ka Gun, Harmonic ka Hai Reciprocal Fun!" 🎉
Mnemonic 2: "D-R-A-M-A" for Arithmetic Sequence Formula 🎭
Use the word D-R-A-M-A to recall the key formulas of AP:
- D = Difference (d)
- R = rth term (ar)
- A = First term (a1)
- M = nth term formula: an = a1 + (n-1)d
- A = Sum of n terms: Sn = n/2 [2a1 + (n-1)d]
Hindi rhyme: "Dekh Rahi ho Arithmetic ka Drama, Formula yaad rakhna hai kamaal ka!" 🎬
Mnemonic 3: "GP Ratio Reminder: 'Raat Ko Ghar'" 🌙🏠
To recall that in GP, the ratio (r) is constant, think:
- Raat (Ratio) ko (constant)
- Ghar (Geometric) mein (sequence)
Meaning: "At night (raat), the ratio stays fixed at home (ghar)" — just like the constant ratio in GP.
Formula reminder: an = a1 × rn-1
Mission: Master This Topic!
Reinforce what you learned with fun activities
Ready to Battle? Test Your Knowledge!
Practice MCQs, build combos, climb the leaderboard!
Start Practice