🌎 Real-World Hook: Have you ever saved pocket money — ₹20 on day 1, ₹20 on day 2, and so on? By day 10 you have ₹200. That steady growth is an arithmetic progression! Or imagine bacteria that double every hour — 1, 2, 4, 8, 16 ... That explosive growth is a geometric progression! Sequences are everywhere: the stitches in knitting, the notes in a musical scale, the floors of a building, the pattern of stadium seats. Once you spot the rule, you can predict any term — even the 100th — without listing them all!
Each step is 4 higher than the last. That makes it an AP!
In an AP, the jumps between terms are always the same size.
In a GP, each stage has double the dots. Growth is very fast!
AP grows by the same amount each time. GP grows by the same factor — much faster!
Write the sum forward and backward. Each pair adds to n+1. There are n pairs. Divide by 2.
An AP has first term a = 5 and common difference d = 4.
Find the 8th term.
t(n) = a + (n − 1) × d
Find 1 + 2 + 3 + ... + 15.
S = n × (n + 1) ÷ 2
A GP has first term a = 3 and common ratio r = 2.
Find the 7th term.
t(n) = a × r(n−1)
| Word / Rule | What It Means |
|---|---|
| Sequence | An ordered list of numbers (terms) |
| Term | One number in a sequence |
| Explicit rule | Formula that uses n to find any term |
| Recursive rule | Uses the previous term to find the next |
| AP (Arithmetic Progression) | Add the same number (d) each time |
| d (common difference) | The fixed amount added in an AP |
| AP nth term | t(n) = a + (n − 1) × d |
| Sum of 1 to n | S = n × (n + 1) ÷ 2 |
| GP (Geometric Progression) | Multiply by the same number (r) each time |
| r (common ratio) | The fixed multiplier in a GP |
| GP nth term | t(n) = a × r(n−1) |
| Finite sequence | Has a last term (ends) |
| Infinite sequence | Goes on forever (no end) |
1. A sequence is an ordered list of ________.
2. In an AP, we add the same number called the ________.
3. The nth term of an AP is t(n) = ________ + (n − 1) × ________.
4. In a GP, we multiply by the same number called the ________.
5. Sum of 1 to n = n × (n + 1) ÷ ________.
1. 2, 5, 8, 11, 14 is an AP with d = 3. ________
2. 1, 3, 9, 27 is an AP. ________
3. In a GP, you add the same number each time. ________
4. The sum of 1 to 10 is 55. ________
1. AP has a = 7, d = 3. Find the 10th term.
2. Find the sum of 1 + 2 + 3 + ... + 20.
1. AP — (a) Multiply by same number
2. GP — (b) Add same number
3. Common difference — (c) n(n+1)/2
4. Sum of 1 to n — (d) The fixed amount added
1. AP: 4, 9, 14, __, __, __
2. GP: 5, 10, 20, __, __, __
💡 Quick check: The sequence is 7, 10, 13, 16, 19, 22, 25, 28, 31, 34. Count to the 10th term to confirm!
💡 Why does this work? Pair up: 1+20=21, 2+19=21, 3+18=21, ... There are 10 such pairs. 10 × 21 = 210.
An ordered list of numbers. Each number is a term.
A formula using n to find any term directly. Example: t(n) = 4n + 1.
A sequence that adds the same number (d) each time. Example: 5, 8, 11, 14.
t(n) = a + (n − 1) × d. a = first term, d = common difference.
S = n × (n + 1) ÷ 2. Example: 1 to 10 = 55.
A sequence that multiplies by the same number (r) each time. Example: 4, 12, 36.
t(n) = a × r(n−1). a = first term, r = common ratio.
AP adds d each time. GP multiplies by r each time. GP grows much faster!
All the key ideas about sequences and progressions in one picture.