Exercise 6.5 — nth Term of GP
General (nth) term of a geometric progression.
Exercise 6.5 — Geometric Progressions (Finding the nth Term)
Exercise 6.5 is part of Chapter 6, Progressions, in Class 10 Mathematics for students following the CBSE, Telangana, and Andhra Pradesh board syllabi. This exercise focuses entirely on Geometric Progressions (GP) — sequences in which every term is obtained by multiplying the previous term by a fixed number called the common ratio.
Unlike an Arithmetic Progression (AP), where consecutive terms differ by a constant added amount, a Geometric Progression grows or shrinks by a constant multiplying factor. This exercise builds your skill in identifying the common ratio, writing the general nth term formula, and solving real exam-style problems involving unknown terms, unknown ratios, and two GPs with a common term.
How the nth Term Formula Is Derived
Before solving problems, it helps to see where the formula aₙ = a·rn−1 actually comes from. Consider the general form of a GP: a, ar, ar², ar³, ar⁴, … Each term is the previous term multiplied by the common ratio r.
Notice the pattern in the exponents: the 1st term has exponent 0, the 2nd term has exponent 1, the 3rd term has exponent 2, and so on. In general, the exponent is always one less than the term number. This pattern generalizes to:
aₙ = a · r^(n−1)This single formula lets you calculate any term of a GP — whether it's the 5th term or the 50th term — as long as you know the first term a and the common ratio r.
Question 1 — Find the Common Ratio r and the nth Term aₙ
In this question, you're given four different geometric progressions and asked to find the common ratio r for each, then write the general nth term formula. The common ratio is always found by dividing any term by the term immediately before it: r = a₂ ÷ a₁.
Quick Answer Summary — Question 1
| Part | G.P. | r | aₙ |
|---|---|---|---|
| (i) | 3, 3/2, 3/4, 3/8, ... | 1/2 | 3/2ⁿ⁻¹ |
| (ii) | 2, −6, 18, −54, ... | −3 | 2(−3)ⁿ⁻¹ |
| (iii) | −1, −3, −9, −27, ... | 3 | −1(3)ⁿ⁻¹ |
| (iv) | 5, 2, 4/5, 8/25, ... | 2/5 | 5(2/5)ⁿ⁻¹ |
Question 2 — Find the 10th and nth Term of the G.P. 5, 25, 125, ...
This question asks for two things: the specific 10th term and the general nth term formula. Both require finding the first term and common ratio first.
Question 3 — Find the Indicated Term of Each Geometric Progression
Here, you are directly given the first term a and the common ratio r, and asked to find a specific term using the nth term formula.
Question 4 — Which Term of the G.P. Equals a Given Value?
This is a reverse problem: instead of finding a term's value, you're given the value and must find which position (n) it occupies in the sequence. The strategy is to set aₙ equal to the given value, substitute the formula, and solve for n by matching exponents.
Question 5 — Find the 12th Term Using the 8th Term and Common Ratio
Given that the 8th term of a GP is 192 and the common ratio is 2, find the 12th term. Instead of finding the first term separately, this question is solved more efficiently using the relationship between terms that are a fixed number of positions apart.
aₘ = aₖ × r^(m−k) (jumping from the kth term to the mth term)Question 6 — Find the GP When Two Terms Are Given
The 4th term of a GP is 2/3, and the 7th term is 16/81. This question asks you to find the complete geometric series — meaning you must determine both the first term a and the common ratio r from two equations.
Question 7 — Two GPs With Equal nth Terms
This is one of the more advanced questions in the exercise. Two separate geometric progressions, 162, 54, 18, ... and 2/81, 2/27, 2/9, ..., are given. We are told their nth terms are equal for some value of n, and we must find that value of n.
Common Mistakes to Avoid
- Computing r incorrectly: Always divide a later term by the term immediately before it (a₂ ÷ a₁), never the reverse. Dividing a₁ ÷ a₂ gives the reciprocal of r and leads to wrong answers throughout the problem.
- Forgetting the (n−1) exponent: A very common error is writing aₙ = a·rⁿ instead of the correct aₙ = a·r^(n−1). Always double-check the exponent is one less than the term number.
- Sign errors with negative common ratios: When r is negative (as in Question 1(ii) and (iii)), be careful with parentheses — (−3)^(n−1) is very different from −3^(n−1). Always keep the negative sign and the base together in brackets.
- Mixing up "jump" formula direction: In problems like Question 5, the relation aₘ = aₖ × r^(m−k) only works when m is greater than k. If asked to go backward (e.g., from a₁₂ to a₈), you would divide by r^(m−k) instead of multiplying.
- Not simplifying powers to a common base: In "which term is this value" problems (Question 4), always express the given value as a power of the same base as the common ratio. Skipping this step makes the equation impossible to solve directly.
Quick Reference — All Answers at a Glance
| Question | Given | Answer |
|---|---|---|
| Q1(i) | 3, 3/2, 3/4, 3/8, ... | r = 1/2, aₙ = 3/2ⁿ⁻¹ |
| Q1(ii) | 2, −6, 18, −54, ... | r = −3, aₙ = 2(−3)ⁿ⁻¹ |
| Q1(iii) | −1, −3, −9, −27, ... | r = 3, aₙ = −1(3)ⁿ⁻¹ |
| Q1(iv) | 5, 2, 4/5, 8/25, ... | r = 2/5, aₙ = 5(2/5)ⁿ⁻¹ |
| Q2 | 5, 25, 125, ... | a₁₀ = 5¹⁰, aₙ = 5ⁿ |
| Q3(i) | a=9, r=1/3, find a₇ | a₇ = 1/81 |
| Q3(ii) | a=−12, r=1/3, find a₆ | a₆ = −4/81 |
| Q4(i) | 2, 8, 32, ... = 512? | 5th term |
| Q4(ii) | 3, 3√3, 9, ... = 729? | 12th term |
| Q4(iii) | 1/3, 1/9, 1/27, ... = 1/2187? | 7th term |
| Q5 | a₈ = 192, r = 2, find a₁₂ | a₁₂ = 3072 |
| Q6 | a₄ = 2/3, a₇ = 16/81 | a = 9/4, r = 2/3 |
| Q7 | Two GPs with equal nth term | n = 5 |
What This Exercise Prepares You For
Exercise 6.5 builds the essential skills needed for the rest of Chapter 6, particularly the upcoming exercises on the sum of n terms of a GP and applications of progressions to real-life problems like compound interest and population growth. A strong grip on the nth term formula here makes those sum formulas much easier to understand and apply.
This exercise also connects naturally to Introduction to Progressions, where the foundational concepts of sequences, arithmetic progressions, and geometric progressions are introduced. For Class 9 students revisiting exponent rules used heavily in this exercise, refer back to Exponents and Powers, since comparing powers with the same base is the central technique in Questions 4 and 7.