Step-by-Step Guide: Formula for the nth Term of a GP
Calculating the nth term of a GP (Geometric Progression) is crucial for many school exams and competitive tests. It helps students solve problems quickly without writing out long sequences. Mastering this saves time and boosts confidence, especially when tackling sums and sequences in maths.
Formula Used in nth term of a GP
The standard formula is: \( a_n = a \times r^{n-1} \), where “a” is the first term, “r” is the common ratio, and “n” is the term position.
Here’s a helpful table to understand nth term of a GP more clearly:
nth term of a GP Table
| Term Position (n) | Formula Used | Sample Value |
|---|---|---|
| 1 (First) | a × r0 | a |
| 2 (Second) | a × r1 | a × r |
| 3 (Third) | a × r2 | a × r² |
| n (Any Term) | a × rn-1 | Depends on n, a, r |
This table shows how the formula for the nth term of a GP applies to any position in the sequence, simply by changing the value of n.
What is a Geometric Progression?
A geometric progression (GP) is a number sequence where each term is found by multiplying the previous term by a common ratio (r). For example, in the sequence 2, 4, 8, 16, ... each term is multiplied by 2 to get the next. The GP formula allows you to directly calculate any term in the sequence without listing all previous terms. To learn more about sequences and related ideas, see sequences and series.
Worked Example – Solving a Problem
1. Write down the problem: Find the 5th term of a GP where the first term (a) is 3 and the common ratio (r) is 2.2. Use the formula: \( a_n = a \times r^{n-1} \)
3. Calculate the exponent: \( 2^{5-1} = 2^{4} = 16 \)
4. Perform the multiplication: \( 3 \times 16 = 48 \)
You can practice more problems or explore the arithmetic-geometric sequence for advanced cases.
Practice Problems
- Find the 7th term of a GP where a = 2 and r = 3.
- In the GP 5, 10, 20, ..., what is the 6th term?
- If the fifth term of a GP is 81 and its first term is 3, with r > 0, find r.
- Which term of the GP 1, 4, 16, ... is 1024?
Common Mistakes to Avoid
- Confusing nth term of a GP with the arithmetic progression (AP) nth term formula (AP uses addition, GP uses multiplication).
- Forgetting that the power in r should always be (n-1), not n.
- Mixing up the order of the sequence (using wrong values for n).
- Using the wrong common ratio (r) by not checking two consecutive terms for multiplication.
Real-World Applications
The concept of nth term of a GP is used in many practical fields, such as calculating compound interest in banking, analyzing population growth, and studying patterns in investments or energy usage. Understanding geometric progressions helps connect these maths concepts to real-world situations that Vedantu explains clearly for students.
We explored the idea of nth term of a GP, its formula, detailed steps to solve related problems, and its importance in real-world settings. Review the formula and keep practicing with Vedantu for exam-ready confidence in sequences and series.
To further deepen your understanding or to tackle competitive exam questions, visit these helpful resources: sum of GP and arithmetic progression for comparisons, and nth term of an AP for AP term calculations.
FAQs on How to Find the nth Term of a Geometric Progression (GP)
1. What is the formula for the nth term of a geometric progression (GP)?
The nth term of a geometric progression (GP) is given by the formula: an = a × rn–1 where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.
2. How do you find the nth term of a geometric sequence?
To find the nth term of a geometric sequence, use this formula: an = a × rn–1, where 'a' is the first term, 'r' is the common ratio, and 'n' is the required term's position.
3. How do you determine the sum of the first n terms of a geometric progression?
The sum of the first n terms of a GP (Sn) can be found using: Sn = a (1 – rn) ⁄ (1 – r), if r ≠ 1. If r = 1, the sum is Sn = n × a.
4. What is the general nth term formula of a GP series?
The general nth term of a GP is an = a × rn–1, where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.
5. How do you find the nth term of a GP from the end?
To find the nth term from the end of a GP with the last term 'l', first term 'a', and total number of terms 'N', use:
Term from end = l × (1/r)n–1
6. What is the nth term of the geometric progression 5, 25, 125,...?
For the sequence 5, 25, 125,..., the first term 'a' = 5 and common ratio 'r' = 5.
The nth term is an = 5 × 5n–1 = 5n.
7. How do you find the common ratio in a geometric progression?
The common ratio (r) of a geometric progression is found by dividing any term by its preceding term: r = a2 ⁄ a1 or r = an ⁄ an–1.
8. What is the formula for the sum to infinity of a GP?
For a GP where the common ratio |r| < 1, the sum to infinity is S∞ = a ⁄ (1 – r), where 'a' is the first term.
9. How can you use the nth term to identify if a sequence is geometric?
A sequence is geometric if every term after the first is obtained by multiplying the previous term by a constant (the common ratio r). Check if an ⁄ an–1 = r for all terms.
10. What is the difference between nth term of an AP and a GP?
An arithmetic progression (AP) has its nth term as an = a + (n–1)d, where 'd' is the common difference. A geometric progression (GP) uses an = a × rn–1, where 'r' is the common ratio.
11. How do you find the last term of a finite GP?
The last term of a finite GP with 'n' terms, first term 'a', and common ratio 'r' is an = a × rn–1.
12. Can you prove the formula for the nth term of a GP?
Yes, the formula an = a × rn–1 can be derived by repeatedly multiplying the first term 'a' by the common ratio 'r', (n–1) times to reach the nth term.
