Ever noticed how the numbers on a staircase seem to climb in a predictable way?
In simple words, a sequence is just a list of numbers written one after another. A series adds those numbers together. Think of a sequence as the steps and a series as the total height you reach after climbing them.
What is a Sequence?
A sequence is an ordered collection of numbers. Each number in the list is called a term. The first term is written as a1, the second as a2, and so on. When we can write a rule that tells us any term directly, we call it the general term, usually written as an where n tells you the position.
Common Types of Sequences in Class 11
Arithmetic Progression (AP)
An arithmetic progression is a sequence where the difference between consecutive terms stays the same. That constant difference is called the common difference (d). For example, 2, 5, 8, 11,… has a common difference of 3.
First term = a, common difference = d, nth term formula: an = a + (n-1)d.
Geometric Progression (GP)
A geometric progression is a sequence where each term is multiplied by the same number to get the next term. That constant multiplier is called the common ratio (r). For example, 3, 6, 12, 24,… has a common ratio of 2.
First term = a, common ratio = r, nth term formula: an = a·rn-1.
Series: Adding Up the Terms
When you add the terms of a sequence, you get a series. The sum of the first n terms of an AP is written as Sn. The formula is:
Sn = n/2 · (2a + (n‑1)d). It comes from pairing the first and last terms, then the second and second‑last, and so on.
For a GP, the sum of the first n terms (when r ≠ 1) is:
Sn = a·(1‑rⁿ) / (1‑r).
Worked Example: Sum of an AP
Find the sum of the first 10 terms of the AP: 4, 9, 14, …
- First term a = 4
- Common difference d = 9‑4 = 5
- Number of terms n = 10
- Use the sum formula: S10 = 10/2 · (2·4 + (10‑1)·5)
- Calculate inside: 2·4 = 8, (10‑1)·5 = 45, so 8+45 = 53
- S10 = 5 · 53 = 265
So the total of the first ten numbers is 265.
Worked Example: Sum of a GP
Find S5 for the GP: 2, 6, 18, … (r = 3)
- a = 2, r = 3, n = 5
- S5 = 2·(1‑3⁵) / (1‑3)
- 3⁵ = 243, so numerator = 2·(1‑243) = 2·(‑242) = ‑484
- Denominator = 1‑3 = ‑2
- S5 = (‑484) / (‑2) = 242
The first five terms add up to 242.
Quick Comparison: AP vs. GP
| Feature | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
| Rule | Add a constant difference (d) each step | Multiply by a constant ratio (r) each step |
| nth term | an = a + (n‑1)d | an = a·rn‑1 |
| Sum formula | Sn = n/2 · (2a + (n‑1)d) | Sn = a·(1‑rⁿ)/(1‑r) (r≠1) |
| Typical use | Linear growth like monthly savings | Exponential growth like bacteria count |
Step‑by‑Step Flowchart: Finding the Sum of an AP
📝 Likely Exam Questions
- Find the 15th term of the AP 7, 12, 17,…
Answer: a = 7, d = 5, a15 = 7 + (15‑1)·5 = 7 + 70 = 77. - Sum the first 8 terms of the GP 5, 15, 45,…
Answer: a = 5, r = 3, n = 8, S8 = 5·(1‑3⁸)/(1‑3) = 5·(1‑6561)/‑2 = 5·6560/2 = 16400. - If the sum of the first n terms of an AP is 210 and a = 5, d = 4, find n.
Answer: Use Sn = n/2 · (2·5 + (n‑1)·4) = 210. Simplify: n(10 + 4n‑4)/2 = 210 → n(4n+6)/2 = 210 → 2n(2n+3) = 210 → n(2n+3)=105. Trying n=7 gives 7·17=119, too high. n=5 gives 5·13=65, low. n=6 gives 6·15=90, low. n=7 gives 119, high. n=... solving quadratic 2n²+3n‑105=0 → n≈7.5 not integer, so maybe mis‑print; typical answer n=10 gives 10·23=230, still off. In exam they'd expect n=9 giving 9·21=189, close. Check work; correct n=10 gives S=10/2·(2·5+9·4)=5·(10+36)=5·46=230, not 210. So answer: n=9 gives S=9/2·(10+32)=4.5·42=189. No integer fits 210, so answer: no integer n satisfies given data. - Show that the sum of the first n natural numbers is n(n+1)/2.
Answer: Write the series 1+2+…+n as S. Reverse it: n+ (n‑1)+…+1 = S. Add both: 2S = (n+1) + (n+1)+… (n terms) = n(n+1). Hence S = n(n+1)/2. - Convert the AP 3, 8, 13,… into a GP by multiplying each term by a constant k. Find k if the first two terms of the GP become 6 and 12.
Answer: AP terms: 3,8,… Multiply by k → 3k = 6 ⇒ k = 2. Check second term: 8·2 = 16, not 12, so no single k works. Therefore such a conversion is impossible.