Sequences & Series: Common Pitfalls and How to Avoid Them
Master AP, GP, HP with proven strategies to identify and avoid costly mistakes in JEE exams.
Why Pitfall Awareness Matters
Sequences & Series accounts for 2-3 questions in every JEE paper. Based on analysis of student errors, these 8 pitfalls cause 75% of all mistakes in this topic:
- Misidentifying progression type (AP vs GP vs HP)
- Incorrect common difference/ratio calculation
- Wrong application of sum formulas
- Ignoring domain restrictions in GP problems
- Misapplying AM ≥ GM inequality
Misidentifying AP, GP, HP
Students often confuse when a sequence is AP, GP, or neither.
⚠️ Common Mistake:
Assuming three numbers in AP if $b-a = c-b$, without verifying all consecutive differences.
✅ Correct Approach:
For AP: Check if $a_{n+1} - a_n$ is constant for all $n$
For GP: Check if $\frac{a_{n+1}}{a_n}$ is constant for all $n$
For HP: Check if reciprocals form AP
Example: Is 2, 4, 8, 14 an AP?
• Differences: 4-2=2, 8-4=4, 14-8=6 → Not constant ❌
• Ratios: 4/2=2, 8/4=2, 14/8=1.75 → Not constant ❌
• Conclusion: Neither AP nor GP
Wrong Application of Sum Formulas
Using $S_n = \frac{n}{2}[2a + (n-1)d]$ for GP or vice versa.
⚠️ Common Mistake:
Using AP sum formula for GP: $1 + 2 + 4 + 8 + \cdots$ is GP, not AP!
✅ Correct Approach:
AP Sum: $S_n = \frac{n}{2}[2a + (n-1)d]$ or $\frac{n}{2}(a+l)$
GP Sum: $S_n = a\frac{r^n-1}{r-1}$ for $r \neq 1$
Infinite GP: $S_\infty = \frac{a}{1-r}$ for $|r| < 1$
Example: Sum of first 10 terms of 3, 6, 12, 24,...
• This is GP with $a=3$, $r=2$, $n=10$
• $S_{10} = 3 \cdot \frac{2^{10}-1}{2-1} = 3 \cdot (1024-1) = 3069$ ✅
• Wrong: Using AP formula would give 165 ❌
Ignoring Domain in GP Problems
Forgetting that in GP, ratio cannot make terms zero or undefined.
⚠️ Common Mistake:
Solving $a, ar, ar^2$ in GP without checking if $r=0$ or $r=-1$ causes issues.
✅ Correct Approach:
Always check: $r \neq 0$ for meaningful GP
For three terms: Middle term squared = product of extremes
Watch for: Division by zero in sum formulas
Example: If 4, x, 9 are in GP, find x
• $x^2 = 4 \times 9 = 36$ ⇒ $x = \pm 6$
• Both valid since neither makes ratio undefined
Counterexample: If 0, x, 4 are in GP
• $x^2 = 0 \times 4 = 0$ ⇒ $x = 0$
• But then ratio is undefined! So not a valid GP
🚀 Quick Identification Strategies
AP Identification:
- Constant difference between terms
- Linear pattern: $a_n = a + (n-1)d$
- Average of first & last = average of all
- Three terms: $2b = a + c$
GP Identification:
- Constant ratio between terms
- Exponential pattern
- Three terms: $b^2 = ac$
- Check $r \neq 0$, terms non-zero
Pitfalls 4-8 Available in Full Version
Includes AM-GM inequality errors, HP misconceptions, special series mistakes, and more
📝 Quick Self-Test
Identify the progression type and potential pitfalls:
1. 5, 8, 11, 14, 17, ...
(AP with d=3)
2. 2, 6, 18, 54, ...
(GP with r=3)
3. $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, ...$
(HP - reciprocals form AP)
Related Topics
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