**CAT Algebra: Progressions & Series – The 99%ile Study Guide**

CAT Algebra: Progressions & Series – The 99%ile Study Guide

What This Is

Progressions and Series (P&S) is a high-frequency, high-scoring topic in CAT QA, appearing in 3–5 questions per paper (including TITA). It tests pattern recognition, algebraic manipulation, and shortcuts—skills that separate 95%ilers from 99%ilers. Mastering P&S ensures quick, accurate solutions (avg. 45–60 sec per question) and frees up time for tougher problems.

Typical CAT Question:
If the sum of the first 10 terms of an AP is 150 and the sum of the next 10 terms is 350, what is the common difference? (A) 1 (B) 2 (C) 3 (D) 4

Key Concepts & Techniques

1. Arithmetic Progression (AP) Basics
2. Formula: ( a_n = a + (n-1)d ), ( S_n = \frac{n}{2}[2a + (n-1)d] )
3. When to use: When terms increase/decrease by a constant difference (e.g., 2, 5, 8, 11…).

4. Shortcut: Sum of first ( n ) terms = ( n \times ) (average of first and last term).

5. Geometric Progression (GP) Basics

6. Formula: ( a_n = ar^{n-1} ), ( S_n = a \frac{1-r^n}{1-r} ) (if ( r \neq 1 ))
7. When to use: When terms multiply by a constant ratio (e.g., 3, 6, 12, 24…).

8. Shortcut: For infinite GP (( |r| < 1 )), ( S_\infty = \frac{a}{1-r} ).

9. Sum of Special Series

10. Sum of first ( n ) natural numbers: ( \frac{n(n+1)}{2} )
11. Sum of first ( n ) squares: ( \frac{n(n+1)(2n+1)}{6} )
12. Sum of first ( n ) cubes: ( \left( \frac{n(n+1)}{2} \right)^2 )

13. When to use: When the series is non-AP/GP but follows a known pattern (e.g., ( 1^2 + 2^2 + \dots + n^2 )).

14. Arithmetic Mean (AM) ≥ Geometric Mean (GM)

15. Formula: ( \frac{a+b}{2} \geq \sqrt{ab} ) (equality iff ( a = b ))

16. When to use: For optimization problems (e.g., "Find the minimum value of ( x + \frac{1}{x} )").

17. Sum of AP Subsets

18. Key Insight: Sum of terms equidistant from start/end is constant.
    • E.g., ( S_{10} = 5(a_1 + a_{10}) = 5(a_2 + a_9) = \dots )

19. When to use: When given partial sums (e.g., sum of first 10 terms vs. next 10 terms).

20. GP Sum Shortcuts

21. For ( r = 1 ): ( S_n = n \times a )
22. For ( r = -1 ): ( S_n = a ) if ( n ) is odd, ( 0 ) if ( n ) is even.

23. When to use: To eliminate options quickly in MCQs.

24. Telescoping Series

25. Concept: Terms cancel out when expanded (e.g., ( \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \dots )).

26. When to use: For fractional series with denominators as products.

27. Recursive Sequences

28. Approach: Find a pattern or express in terms of previous terms.
29. When to use: When the series is defined recursively (e.g., ( a_{n+1} = 2a_n + 1 )).

Step-by-Step Strategy

Follow this process for every P&S question:

1. Identify the Type
2. Is it an AP, GP, or special series? Check for constant difference/ratio.

3. If not, look for patterns (squares, cubes, telescoping, etc.).

4. Write Down Known Formulas

5. Jot down the relevant formula(s) before solving to avoid confusion.

6. Assign Variables

7. For AP: ( a ) (first term), ( d ) (common difference).
8. For GP: ( a ) (first term), ( r ) (common ratio).

9. For special series: ( n ) (number of terms).

10. Set Up Equations

11. Translate the problem into 1–2 equations using the formulas.

12. Example: "Sum of first 10 terms = 150" → ( \frac{10}{2}[2a + 9d] = 150 ).

13. Solve for the Unknown

14. Use substitution or elimination to find the required variable.

15. For MCQs, plug in options to verify.

16. Verify the Answer

17. Check if the answer makes sense (e.g., common difference can’t be negative if terms are increasing).
18. For TITA, recalculate if time permits.

Fully Worked CAT-Style Example

Question:
The sum of the first 10 terms of an AP is 150, and the sum of the next 10 terms is 350. What is the common difference? (A) 1 (B) 2 (C) 3 (D) 4

Solution Using the Strategy:

1. Identify the Type: AP (constant difference).
2. Write Down Formulas:
3. ( S_n = \frac{n}{2}[2a + (n-1)d] )
4. Assign Variables:
5. ( a ) = first term, ( d ) = common difference.
6. Set Up Equations:
7. Sum of first 10 terms: ( \frac{10}{2}[2a + 9d] = 150 ) → ( 5(2a + 9d) = 150 ) → ( 2a + 9d = 30 ) …(1)
8. Sum of next 10 terms (terms 11 to 20):
   • ( S_{20} - S_{10} = 350 )
   • ( \frac{20}{2}[2a + 19d] - 150 = 350 ) → ( 10(2a + 19d) = 500 ) → ( 2a + 19d = 50 ) …(2)
9. Solve for ( d ):
10. Subtract (1) from (2): ( (2a + 19d) - (2a + 9d) = 50 - 30 ) → ( 10d = 20 ) → ( d = 2 ).
11. Verify:
12. Plug ( d = 2 ) into (1): ( 2a + 18 = 30 ) → ( a = 6 ).
13. Check ( S_{20} = 10(12 + 38) = 500 ), which matches ( 150 + 350 = 500 ).

Answer: (B) 2

Common Mistakes

1. Mistake: Assuming all series are AP/GP.
2. Why it happens: Over-reliance on standard formulas.

3. Correct approach: Check for patterns (e.g., squares, cubes) before applying AP/GP.

4. Mistake: Misapplying the sum formula for partial terms.

5. Why it happens: Confusing ( S_{20} - S_{10} ) with ( S_{10} ) of a new AP.

6. Correct approach: For terms 11 to 20, use ( S_{20} - S_{10} ), not a new AP starting at ( a_{11} ).

7. Mistake: Ignoring the condition ( |r| < 1 ) for infinite GP.

8. Why it happens: Forgetting the convergence requirement.

9. Correct approach: Only use ( S_\infty = \frac{a}{1-r} ) if ( |r| < 1 ).

10. Mistake: Arithmetic errors in long calculations.

11. Why it happens: Rushing through steps.

12. Correct approach: Double-check each step (e.g., ( 2a + 9d = 30 ), not ( 2a + 9d = 15 )).

13. Mistake: Not using option elimination in MCQs.

14. Why it happens: Solving from scratch even when options are given.
15. Correct approach: Plug in options to save time (e.g., for ( d = 2 ), check if ( 2a + 9d = 30 )).

CAT Traps & Time Management

1. Trap: Hidden Patterns
2. Example: A series like ( 1, 2, 4, 8, 16 ) is a GP, but ( 1, 2, 4, 7, 11 ) is not (differences are +1, +2, +3…).

3. How to avoid: Always check differences/ratios before assuming AP/GP.

4. Trap: Partial Sums

5. Example: "Sum of terms 6 to 15" is not ( S_{15} - S_5 ) (it’s ( S_{15} - S_5 )).

6. How to avoid: Count terms carefully (terms 6 to 15 = 10 terms).

7. Trap: Negative Common Difference

8. Example: An AP with ( d = -2 ) (e.g., 10, 8, 6, 4…).

9. How to avoid: Never assume ( d ) is positive unless specified.

10. Time Management:

11. AP/GP questions: 45–60 sec.
12. Special series: 60–90 sec (longer if pattern isn’t obvious).
13. TITA questions: 90 sec max (skip if stuck).

Quick Practice

1. Question:
   The sum of the first 5 terms of a GP is 31, and the sum of the next 5 terms is 992. If the common ratio is an integer, what is its value?
   (A) 2 (B) 3 (C) 4 (D) 5

Answer: (A) 2
Solution Path:
( S_5 = a \frac{r^5 - 1}{r - 1} = 31 ), ( S_{10} - S_5 = 992 ) → ( S_{10} = 1023 ).
( \frac{S_{10}}{S_5} = r^5 = 33 ). Only ( r = 2 ) satisfies ( r^5 = 32 ).

1. Question:
   Find the sum of the series ( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots ) to infinity.
   (A) 1 (B) 2 (C) 3 (D) 4

Answer: (B) 2
Solution Path:
Infinite GP with ( a = 1 ), ( r = \frac{1}{2} ). ( S_\infty = \frac{1}{1 - \frac{1}{2}} = 2 ).

Last-Minute Cram Sheet

1. AP Sum: ( S_n = \frac{n}{2}[2a + (n-1)d] ) or ( n \times ) (avg. of first/last term).
2. GP Sum: ( S_n = a \frac{1-r^n}{1-r} ) (if ( r \neq 1 )).
3. Infinite GP Sum: ( S_\infty = \frac{a}{1-r} ) (only if ( |r| < 1 )).
4. Sum of first ( n ) natural numbers: ( \frac{n(n+1)}{2} ).
5. Sum of first ( n ) squares: ( \frac{n(n+1)(2n+1)}{6} ).
6. Sum of first ( n ) cubes: ( \left( \frac{n(n+1)}{2} \right)^2 ).
7. AM ≥ GM: ( \frac{a+b}{2} \geq \sqrt{ab} ) (equality iff ( a = b )).
8. Telescoping Series: Look for terms like ( \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} ).
9. Partial Sums: ( S_{m \text{ to } n} = S_n - S_{m-1} ).
10. ⚠️ Trap: If a series is not AP/GP, check for squares, cubes, or recursive patterns before giving up.

Final Tip: For TITA questions, always recheck calculations—a single arithmetic error can cost you the question!