Progressions – Arithmetic Progression (AP) & Geometric Progression (GP)

1. Introduction

Progressions are sequences of numbers that follow a specific pattern.
They are frequently asked in Quantitative Aptitude sections of SSC, Banking, Railways, and other competitive exams.

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2. Arithmetic Progression (AP)

Definition

A sequence of numbers in which the difference between consecutive terms is constant.

• First term: $a$a
• Common difference: $d$d
• nth term formula:

$$T_n = a + (n-1)d$$Tn​=a+(n−1)d

• Sum of first n terms:

$$S_n = \frac{n}{2} [2a + (n-1)d] \quad \text{or} \quad S_n = \frac{n}{2} (a + l)$$Sn​=2n​[2a+(n−1)d]orSn​=2n​(a+l)

where $l$l is the last term.

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3. Geometric Progression (GP)

Definition

A sequence of numbers in which the ratio of consecutive terms is constant.

• First term: $a$a
• Common ratio: $r$r
• nth term formula:

$$T_n = a \cdot r^{n-1}$$Tn​=a⋅rn−1

• Sum of first n terms (if $r \neq 1$r=1):

$$S_n = a \frac{r^n – 1}{r – 1}$$Sn​=ar−1rn−1​

• Sum to infinity (if $|r|<1$∣r∣<1):

$$S_\infty = \frac{a}{1 – r}$$S∞​=1−ra​

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4. Important Tips

• For AP, always identify first term (a) and common difference (d)
• For GP, always identify first term (a) and common ratio (r)
• Check sum formulas carefully; common mistakes involve signs and powers
• Use nth term formulas for solving missing term problems

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Top 25 Practice Questions – Progressions

Arithmetic Progression (AP)

Q1. Find the 10th term of AP: 2, 5, 8, …
Q2. Sum of first 20 terms of AP: 3, 7, 11, …
Q3. If the 5th term of AP is 12 and the 10th term is 27, find a and d.
Q4. Find the 15th term of AP: 7, 12, 17, …
Q5. Sum of first 50 natural numbers.
Q6. Find the sum of first 25 terms of AP: 4, 9, 14, …
Q7. If a = 3 and d = 5, find 20th term.
Q8. The sum of first n terms is 210. If a = 5 and d = 7, find n.
Q9. Find the 12th term of AP: 10, 7, 4, …
Q10. Sum of first 30 terms of AP: 5, 10, 15, …

Geometric Progression (GP)

Q11. Find the 6th term of GP: 3, 6, 12, …
Q12. Sum of first 8 terms of GP: 2, 4, 8, …
Q13. If the 3rd term is 24 and the 6th term is 192, find a and r.
Q14. Sum to infinity of GP: 5, 10, 20, …
Q15. Find the 10th term of GP: 1, 2, 4, 8, …
Q16. Sum of first 5 terms of GP: 2, 6, 18, …
Q17. Find the 7th term of GP: 81, 27, 9, …
Q18. Sum to infinity of GP: 8, 4, 2, 1, …
Q19. If a = 5 and r = 2, find 8th term.
Q20. Sum of first 10 terms of GP: 1, 3, 9, …
Q21. Find 5th term of GP: 16, 8, 4, …
Q22. Sum of first n terms = 364, a = 1, r = 2. Find n.
Q23. Find the 4th term of GP: 7, 21, 63, …
Q24. If sum to infinity = 15, first term = 10, find r.
Q25. Find sum of first 6 terms of GP: 3, 9, 27, …

Answer

Answers – Progressions

AP Answers

Q1. 26
Q2. 880
Q3. a = 3, d = 3
Q4. 87
Q5. 1275
Q6. 625
Q7. 98
Q8. n = 10
Q9. -20
Q10. 465

GP Answers

Q11. 192
Q12. 510
Q13. a = 8, r = 2
Q14. Infinity sum = ∞ (since r > 1, diverges)
Q15. 512
Q16. 242
Q17. 1
Q18. 15
Q19. 640
Q20. 88573
Q21. 1
Q22. n = 9
Q23. 189
Q24. r = 0.5
Q25. 729