Chapter 6 – Sequences and Series (JEE)
1. What is a Sequence?
A sequence is an ordered list of numbers arranged according to a definite rule.
Order is very important in sequences.
Example: $1,2,3$ is different from $3,2,1$.
2. What is a Series?
A series is the sum of the terms of a sequence.
If a sequence is $a_1, a_2, a_3, \dots$
then series is $a_1 + a_2 + a_3 + \dots$
3. Arithmetic Progression (AP)
A sequence is called an Arithmetic Progression if the difference between
consecutive terms is constant.
$a, a+d, a+2d, a+3d, \dots$
4. General Term of AP
$n^{th}$ term:
$$a_n = a + (n-1)d$$
5. Sum of First n Terms of AP
$$S_n = \frac{n}{2}[2a + (n-1)d]$$
OR
$$S_n = \frac{n}{2}(a + l)$$
6. Arithmetic Mean (AM)
Arithmetic mean between two numbers $a$ and $b$ is:
$$AM = \frac{a+b}{2}$$
7. Properties of AP
- Sum of equidistant terms from start and end is same
- If three numbers are in AP, middle one is their AM
8. Geometric Progression (GP)
A sequence is a GP if the ratio between consecutive terms is constant.
$a, ar, ar^2, ar^3, \dots$
9. General Term of GP
$n^{th}$ term:
$$a_n = ar^{n-1}$$
10. Sum of First n Terms of GP
If $r \ne 1$:
$$S_n = \frac{a(r^n - 1)}{r - 1}$$
11. Sum of Infinite GP
If $|r| < 1$, infinite GP has a finite sum.
$$S_\infty = \frac{a}{1-r}$$
12. Geometric Mean (GM)
Geometric mean between $a$ and $b$:
$$GM = \sqrt{ab}$$
13. Harmonic Progression (HP)
A sequence is HP if the reciprocals of its terms form an AP.
If $a,b,c$ are in HP then $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in AP.
14. Harmonic Mean (HM)
$$HM = \frac{2ab}{a+b}$$
15. Relationship Between AM, GM and HM
$$AM \ge GM \ge HM$$
Equality holds only when $a=b$.
16. Special Series (Very Important)
$$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$$
$$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$
$$\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$$
17. Sigma Notation
Sigma ($\sum$) is used to represent sum of terms compactly.
$$\sum_{k=1}^{n} (2k+1)$$
18. Inserting Arithmetic Means
To insert $n$ arithmetic means between $a$ and $b$:
Common difference:
$$d = \frac{b-a}{n+1}$$
19. Inserting Geometric Means
Common ratio:
$$r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$$
20. Common JEE Traps
- Confusing AP and GP formulas
- Forgetting condition $|r|<1$ for infinite GP
- Wrong identification of first term
- Errors in special series formulas
21. Typical JEE Question Patterns
| Type | Approach |
|---|---|
| Find $n^{th}$ term | Use general term formula |
| Find sum | Use $S_n$ formulas |
| Means problems | Use AM/GM/HM relations |
| Infinite series | Check $|r|<1$ |
22. Final Revision Checklist
You have mastered this chapter if you can:
- Identify AP, GP, HP instantly
- Apply sum formulas correctly
- Solve infinite series confidently
- Use special series without mistakes
- Handle AM–GM–HM relations