Physics and Mathematics
Example – Geometric Progression
Practice Questions: Finding the nᵗʰ Term of a G.P.
Question 1. Find the 7ᵗʰ term of the series 2, 4, 8, 16, 32, …
Step-by-Step Solution:
First term: [a = 2]
Common ratio:
[r = \dfrac{4}{2} = 2]
Formula for nᵗʰ term:
[Tₙ = a r^{n−1}]
Substitute [n = 7]:
[T₇ = 2 × 2⁶]
Calculate power:
[2⁶ = 64]
Multiply:
[T₇ = 128]
Final Answer: 128
Question 2. Find the 8ᵗʰ term of the series 3, 6, 12, 24, …
Step-by-Step Solution:
[a = 3]
[r = \dfrac{6}{3} = 2]
Formula: [Tₙ = a r^{n−1}]
[T₈ = 3 × 2⁷]
[2⁷ = 128]
[T₈ = 384]
Final Answer: 384
Question 3. Find the 6ᵗʰ term of the series 5, 10, 20, 40, …
Step-by-Step Solution:
[a = 5]
[r = \dfrac{10}{5} = 2]
[T₆ = 5 × 2⁵]
[2⁵ = 32]
[T₆ = 160]
Final Answer: 160
Question 4. Find the 9ᵗʰ term of the series 1, 2, 4, 8, …
Step-by-Step Solution:
[a = 1]
[r = 2]
[T₉ = 1 × 2⁸]
[2⁸ = 256]
Final Answer: 256
Question 5. Find the 7ᵗʰ term of the series 81, 27, 9, 3, …
Step-by-Step Solution:
[a = 81]
[r = \dfrac{27}{81} = \dfrac{1}{3}]
[T₇ = 81 × (\dfrac{1}{3})⁶]
[(\dfrac{1}{3})⁶ = \dfrac{1}{729}]
[T₇ = \dfrac{81}{729}]
Simplify:
[T₇ = \dfrac{1}{9}]
Final Answer: 1/9
Question 6. Find the 5ᵗʰ term of the series 7, 14, 28, …
Step-by-Step Solution:
[a = 7]
[r = \dfrac{14}{7} = 2]
[T₅ = 7 × 2⁴]
[2⁴ = 16]
[T₅ = 112]
Final Answer: 112
Question 7. Find the 6ᵗʰ term of the series −2, −6, −18, …
Step-by-Step Solution:
[a = −2]
[r = \dfrac{−6}{−2} = 3]
[T₆ = −2 × 3⁵]
[3⁵ = 243]
[T₆ = −486]
Final Answer: −486
Question 8. Find the 7ᵗʰ term of the series 4, −8, 16, −32, …
Step-by-Step Solution:
[a = 4]
[r = \dfrac{−8}{4} = −2]
[T₇ = 4 × (−2)⁶]
Since power is even:
[(−2)⁶ = 64]
[T₇ = 256]
Final Answer: 256
Question 9. Find the 10ᵗʰ term of the series 1, 3, 9, 27, …
Step-by-Step Solution:
[a = 1]
[r = 3]
[T₁₀ = 1 × 3⁹]
[3⁹ = 19683]
Final Answer: 19683
Question 10. Find the 8ᵗʰ term of the series 6, 18, 54, …
Step-by-Step Solution:
[a = 6]
[r = \dfrac{18}{6} = 3]
[T₈ = 6 × 3⁷]
[3⁷ = 2187]
[T₈ = 13122]
Final Answer: 13122
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