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Kumar Rohan

Physics and Mathematics

Geometric Progression and its nth Term

1. What is a Geometric Progression (G.P.)?

A sequence of numbers is called a Geometric Progression if each term after the first is obtained by multiplying the previous term by a fixed non-zero number.

That fixed number is called the common ratio.

Examples of G.P.

[2, 6, 18, 54, …]
(Each term is multiplied by 3)
[5, 10, 20, 40, …]
(Each term is multiplied by 2)
[81, 27, 9, 3, …]
(Each term is multiplied by [\dfrac{1}{3}])
[1, −2, 4, −8, …]
(Each term is multiplied by −2)

2. Terms and Notations Used in G.P.

Symbol	Meaning
[a]	First term
[r]	Common ratio
[n]	Number of terms
[Tₙ]	nᵗʰ term

3. Common Ratio (r)

The common ratio is defined as:

[ r = \dfrac{\text{Second term}}{\text{First term}} ]

[ r = \dfrac{T₂}{T₁} ]

Important:

[r] must be non-zero
[r] can be positive, negative, or fractional

Example

For the G.P. [3, 9, 27, …]:

[ r = \dfrac{9}{3} = 3 ]

4. nᵗʰ Term of a Geometric Progression

The formula for the nᵗʰ term of a G.P. is:

[ T_{n} = a r^{n−1} ]

Why this formula works (Conceptual Explanation)

First term: [T_1 = a]
Second term: [T_2 = ar]
Third term: [T_3 = ar²]
Fourth term: [T_4 = ar³]

So, the exponent of [r] increases by 1 each time.

Thus, the [n^{th}] term becomes:

[ T_{n} = a r^{n−1} ]

5. Examples with Solutions

Example 1

Find the 6ᵗʰ term of the G.P. [2, 6, 18, …].

Step-by-Step Solution:

Identify the first term:
[a = 2]
Find the common ratio:
[r = \dfrac{6}{2} = 3]
Use the nᵗʰ term formula:
[Tₙ = a r^{n−1}]
Substitute values:
[T₆ = 2 × 3^{5}]
Evaluate power:
[3⁵ = 243]
Multiply:
[T₆ = 486]

Final Answer:
The 6ᵗʰ term is 486.

Example 2

Find the 8ᵗʰ term of the G.P. [5, 10, 20, …].

Step-by-Step Solution:

First term: [a = 5]
Common ratio: [r = \dfrac{10}{5} = 2]
Formula: [Tₙ = a r^{n−1}]
Substitute: [T₈ = 5 × 2^{7}]
Evaluate: [2⁷ = 128]
Multiply: [T₈ = 640]

Final Answer: 640

Example 3

Find the 5ᵗʰ term of the G.P. [81, 27, 9, …].

Step-by-Step Solution:

First term: [a = 81]
Common ratio:
[r = \dfrac{27}{81} = \dfrac{1}{3}]
Formula: [Tₙ = a r^{n−1}]
Substitute:
[T₅ = 81 × (\dfrac{1}{3})⁴]
Simplify power:
[(\dfrac{1}{3})⁴ = \dfrac{1}{81}]
Multiply:
[T₅ = 1]

Final Answer: 1

Example 4

Find the 7ᵗʰ term of the G.P. [1, −2, 4, −8, …].

Step-by-Step Solution:

[a = 1]
[r = −2]
[T₇ = 1 × (−2)⁶]
Since power is even:
[(−2)⁶ = 64]

Final Answer: 64

Example 5

If [a = 3] and [r = 2], find [T₁₀].

Step-by-Step Solution:

Formula: [Tₙ = a r^{n−1}]
Substitute:
[T₁₀ = 3 × 2⁹]
[2⁹ = 512]
Multiply:
[T₁₀ = 1536]

Final Answer: 1536

6. Conceptual Questions with Solutions

1. What is a Geometric Progression?

A Geometric Progression (G.P.) is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed non-zero number called the common ratio. This means the ratio of any term to its preceding term remains constant throughout the sequence.

2. How is a G.P. different from an A.P.?

In a G.P., the ratio between consecutive terms is constant, whereas in an A.P., the difference between consecutive terms is constant. So, G.P. is based on multiplication, while A.P. is based on addition or subtraction.

3. What is meant by the common ratio?

The common ratio is the constant number obtained by dividing any term of a G.P. by its preceding term. Mathematically, [ r = \dfrac{T₂}{T₁} ]. It determines how fast the terms grow, decrease, or alternate.

4. Can the common ratio be negative?

Yes, the common ratio can be negative. When the common ratio is negative, the terms of the G.P. alternate in sign, meaning positive and negative terms appear alternately.

5. Can the common ratio be a fraction?

Yes. If the common ratio is a fraction such that [|r| < 1], the terms of the G.P. gradually decrease in magnitude. Such G.P.s are very important in topics like infinite geometric series.

6. What happens if the common ratio is 1?

If [r = 1], then every term of the G.P. is equal to the first term. So the sequence becomes a constant sequence like [5, 5, 5, 5, …].

7. Why must the common ratio be non-zero?

If the common ratio were zero, division by zero would be involved while finding [r], which is undefined. Hence, a valid G.P. must always have a non-zero common ratio.

8. What is the general form of a G.P.?

The general form of a G.P. is: [a, ar, ar², ar³, …] where a is the first term and r is the common ratio.

9. What is the formula for the nᵗʰ term of a G.P.?

The nᵗʰ term of a G.P. is given by: [ Tₙ = a r^{n−1} ] This formula helps us find any term directly without writing all previous terms.

10. Why does the exponent of r become (n−1)?

The first term has no multiplication by [r], the second term has one multiplication, the third has two, and so on. Thus, the nᵗʰ term has [n−1] multiplications of [r].

11. Is every increasing sequence a G.P.?

No. For a sequence to be a G.P., the ratio between consecutive terms must be constant, not just increasing. Many increasing sequences are not G.P.s.

12. Can a G.P. have decreasing terms?

Yes. If the common ratio satisfies [0 < r < 1], the G.P. will have decreasing terms.

13. Can a G.P. contain both positive and negative terms?

Yes. This happens when the common ratio is negative, causing alternate signs.

14. Is it necessary to know all terms to find the nᵗʰ term?

No. Knowing just the first term and the common ratio is sufficient to find any term.

15. Where are G.P.s used in real life?

G.P.s are used in compound interest, population growth, radioactive decay, and many physics and finance applications.

7. FAQs / Common Misconceptions

1. A G.P. is identified by constant difference.

This is incorrect. A G.P. is identified by a constant ratio, not a constant difference.

2. Common ratio must always be greater than 1.

Incorrect. The common ratio can be less than 1, greater than 1, or even negative.

3. Negative common ratio means decreasing G.P.

Wrong. A negative common ratio causes alternating signs, not necessarily decrease.

4. Fractional common ratio does not form a G.P.

Incorrect. Fractional values of [r] form valid G.P.s.

5. The nᵗʰ term formula works only for positive r.

False. The formula works for all non-zero values of [r].

6. Every sequence with multiplication is a G.P.

Incorrect. The multiplication factor must be constant.

7. If r = 1, the sequence is not a G.P.

Wrong. It is a valid G.P. with all terms equal.

8. Zero can be the common ratio.

Incorrect. Division by zero is undefined, so [r ≠ 0].

9. A G.P. must always increase.

False. A G.P. can increase, decrease, or alternate.

10. G.P. and exponential functions are unrelated.

Incorrect. G.P.s are the discrete form of exponential growth and decay.

8. Practice Questions with Very Detailed Step-by-Step Solutions

Practice Question 1. Find the 9ᵗʰ term of the G.P. [3, 6, 12, …].

Step-by-Step Solution:

[a = 3]

[r = \dfrac{6}{3} = 2]

Formula: [Tₙ = a r^{n−1}]

[T₉ = 3 × 2⁸]

[2⁸ = 256]

[T₉ = 768]

Answer: 768

Practice Question 2. Find the 6ᵗʰ term of the G.P. [16, 8, 4, …].

Step-by-Step Solution:

[a = 16]

[r = \dfrac{8}{16} = \dfrac{1}{2}]

[T₆ = 16 × (\dfrac{1}{2})⁵]

[(\dfrac{1}{2})⁵ = \dfrac{1}{32}]

[T₆ = \dfrac{16}{32} = \dfrac{1}{2}]

Answer: 1/2

Practice Question 3. Find the 7ᵗʰ term of the G.P. [−2, 6, −18, …].

Step-by-Step Solution:

[a = −2]

[r = \dfrac{6}{−2} = −3]

[T₇ = −2 × (−3)⁶]

[(−3)⁶ = 729]

[T₇ = −1458]

Answer: −1458

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