1st Year Math Chapter 6 Exercise 6.8 Notes: Sum of a Geometric Series — Finite and Infinite (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Exercise 6.8 mirrors Exercise 6.4, but for geometric progressions — covering the sum of a finite number of terms of a G.P., and the special case of summing infinitely many terms.
What Does This Exercise Cover?
This exercise teaches the formula for adding up n terms of a geometric series, and introduces the important condition under which an infinite geometric series actually has a finite total.
Key Concepts
1. Sum of n Terms of a Finite G.P.
The sum of the first n terms of a G.P. (where r ≠ 1) is given by Sₙ = a₁(1 − rⁿ) / (1 − r).
2. Sum of an Infinite Geometric Series
When |r| < 1, the terms of a G.P. shrink toward zero as n grows, allowing the series to approach a finite total: S∞ = a₁ / (1 − r).
3. When an Infinite Series Has No Sum
If |r| ≥ 1, the terms do not shrink toward zero, and the infinite series does not settle on a finite total — it has no sum.
Sum Formulas for a Geometric Series
| Formula | Condition |
| Sₙ = a₁(1 − rⁿ) / (1 − r) | Finite sum of n terms, for any r ≠ 1 |
| S∞ = a₁ / (1 − r) | Infinite sum, valid only when |r| < 1 |
Solved Examples
Example 1: Find the sum of the first 6 terms of the G.P. 3, 6, 12, 24, …
a₁ = 3, r = 2, n = 6.
Sₙ = 3(1 − 2⁶)/(1 − 2) = 3(1 − 64)/(−1) = 3(−63)/(−1).
Answer: S₆ = 189
Example 2: Find the sum to infinity of the G.P. 8, 4, 2, 1, …
a₁ = 8, r = 1/2.
Since |r| < 1, S∞ = 8 / (1 − 1/2) = 8 / (1/2).
Answer: S∞ = 16
Sample MCQs
1. The formula for the sum of n terms of a G.P. (r ≠ 1) is:
a) Sₙ = a₁(1 − rⁿ)/(1 − r) b) Sₙ = (n/2)(a₁ + aₙ) c) Sₙ = a₁ × rⁿ d) Sₙ = n × a₁
Answer: a) Sₙ = a₁(1 − rⁿ)/(1 − r)
2. An infinite geometric series has a finite sum only when:
a) r = 1 b) |r| < 1 c) r > 1 d) r = 0
Answer: b) |r| < 1
3. The formula for the sum to infinity of a G.P. is:
a) S∞ = a₁ × r b) S∞ = a₁ / (1 − r) c) S∞ = a₁ − r d) S∞ = a₁ / (1 + r)
Answer: b) S∞ = a₁ / (1 − r)
4. The sum to infinity of 4, 2, 1, 0.5, … is:
a) 4 b) 6 c) 8 d) 10
Answer: c) 8
5. If |r| ≥ 1 in a geometric series, the infinite sum:
a) Equals zero b) Does not exist (is undefined) c) Equals a₁ d) Equals r
Answer: b) Does not exist (is undefined)
Important Short Questions
- State the formula for the sum of the first n terms of a G.P.
- State the condition required for an infinite geometric series to have a finite sum.
- Find the sum of the first 5 terms of the G.P. 2, 6, 18, …
- Find the sum to infinity of the G.P. 9, 3, 1, …
- Explain why a G.P. with r = 2 cannot have a finite sum to infinity.
Important Long Questions
- Find the sum of the first 7 terms of the G.P. 1, 3, 9, 27, …
- Find the sum to infinity of the G.P. 12, 6, 3, 1.5, …, showing that the condition |r| < 1 is satisfied.
- The sum to infinity of a G.P. is 20, and its first term is 5. Find the common ratio.
- Explain, using an example, why the terms of a geometric series must shrink toward zero for an infinite sum to exist.
How to Approach This Exercise Effectively
- Always check the value of r before choosing a formula — a finite sum works for any r ≠ 1, but the infinite sum formula strictly requires |r| < 1.
- If a question asks for an infinite sum but doesn’t confirm |r| < 1, check this condition yourself before applying the formula.
- Memorize both sum formulas side-by-side; they look similar but serve very different purposes.
- For ‘find r’ problems using the infinite sum, rearrange S∞ = a₁/(1−r) algebraically before substituting known values.
- Practice recognizing G.P.s written with fractions or decimals as the common ratio, since infinite sum problems often use these.
FAQs
Q: Why doesn’t the infinite sum formula work for every G.P.?
A: Because if |r| ≥ 1, the terms don’t shrink toward zero, so adding infinitely many of them never settles on a finite value.
Q: What’s the difference between a finite sum and an infinite sum of a G.P.?
A: A finite sum adds up a specific number of terms (n terms), while an infinite sum considers what the total approaches as more and more terms are added forever.
Q: Can the common ratio be negative in an infinite geometric series?
A: Yes — as long as |r| < 1, a negative ratio (like −0.5) still gives a valid, finite infinite sum.
Q: What real-world situations use the sum of an infinite geometric series?
A: Examples include the total distance traveled by a bouncing ball that loses height each bounce, or the total value of a repeating decimal expressed as a fraction.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.8.