Ever noticed how the numbers in a video game’s high‑score list seem to follow a pattern? That hidden pattern is exactly what sequences and series are all about.
💡 In Simple Words: A sequence is a list of numbers written in a specific order. When you add up the numbers of a sequence, you get a series. Think of a sequence as a train of cars and a series as the total weight of all the cars together.
What is a Sequence?
A sequence is just a set of numbers arranged one after another. Each number is called a term (the first term, second term, and so on). If you can predict the next term using a rule, the sequence is called deterministic.
Arithmetic Progression (AP)
An arithmetic progression is a sequence where each term after the first is obtained by adding the same number, called the common difference (denoted d), to the previous term. It’s like climbing stairs that are all the same height.
- First term = a
- n‑th term = a + (n‑1)d
Example: 3, 7, 11, 15, … Here a = 3 and d = 4. The 5th term is 3 + (5‑1)·4 = 19.
Geometric Progression (GP)
A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by the same number, called the common ratio (denoted r). Imagine a balloon that inflates by the same factor each second.
- First term = a
- n‑th term = a·r^(n‑1)
Example: 2, 6, 18, 54, … Here a = 2 and r = 3. The 4th term is 2·3^(4‑1) = 54.
From Sequence to Series
When you add the terms of a sequence, you form a series. The sum of the first n terms of an AP is called the arithmetical series, and the sum of the first n terms of a GP is the geometrical series.
Sum of an AP
The formula S_n = n/2·(2a + (n‑1)d) gives the sum of the first n terms. It’s like pairing the first and last term, the second and second‑last, and so on – each pair adds up to the same total.
Worked Example: Find the sum of the first 10 terms of 5, 9, 13, …
Here a = 5, d = 4, n = 10.
S_10 = 10/2·[2·5 + (10‑1)·4] = 5·[10 + 36] = 5·46 = 230.
Sum of a GP
If r ≠ 1, the sum S_n = a·(1‑r^n)/(1‑r). If r = 1, the series is just n·a because every term is the same.
Worked Example: Find the sum of the first 5 terms of 3, 12, 48, …
a = 3, r = 4, n = 5.
S_5 = 3·(1‑4^5)/(1‑4) = 3·(1‑1024)/(‑3) = 3·(‑1023)/(‑3) = 3·341 = 1023.
Key Formulas at a Glance
| Concept | Formula |
|---|---|
| n‑th term of AP | a + (n‑1)d |
| Sum of n terms of AP | n/2·[2a + (n‑1)d] |
| n‑th term of GP | a·r^(n‑1) |
| Sum of n terms of GP (r≠1) | a·(1‑r^n)/(1‑r) |
Common Mistakes to Avoid
- Mixing up the common difference (d) with the common ratio (r). Remember, d is added, r is multiplied.
- For GP sum, forgetting to change the sign when r is greater than 1 (the denominator becomes negative).
- Using the AP sum formula for a GP – the patterns are different.
📝 Likely Exam Questions
- Find the 12th term of the AP: 4, 9, 14, …
Solution: a = 4, d = 5, n = 12 → T_12 = 4 + (12‑1)·5 = 4 + 55 = 59. - Sum the first 7 terms of the GP: 2, 6, 18, …
Solution: a = 2, r = 3, n = 7 → S_7 = 2·(1‑3^7)/(1‑3) = 2·(1‑2187)/(‑2) = 2·(‑2186)/(‑2) = 2·1093 = 2186. - If the 3rd term of an AP is 15 and the 7th term is 27, find the sum of the first 10 terms.
Solution: Let a be first term, d common difference.
T_3 = a + 2d = 15
T_7 = a + 6d = 27
Subtract: 4d = 12 → d = 3.
Plug back: a + 2·3 = 15 → a = 9.
Now S_10 = 10/2·[2·9 + (10‑1)·3] = 5·[18 + 27] = 5·45 = 225. - Determine whether the series 5, 10, 20, 40, … is arithmetic or geometric and find its 6th term.
Solution: Ratios 10/5 = 2, 20/10 = 2, 40/20 = 2 → constant ratio → GP.
a = 5, r = 2, n = 6 → T_6 = 5·2^(5) = 5·32 = 160.