If the 3rd and 7th terms of a G.P are 9 and 1/9 respectively. Find the common ratio.

Analyze the question: We have a geometric progression (G.P.) where the 3rd term is 9 and the 7th term is 1/9. We need the common ratio r.

Concept: The nth term of a G.P. is T_n = ar^(n-1).

Solution:

T₃ = ar² = 9 T₇ = ar⁶ = 1/9

Divide T₇ by T₃:

ar⁶ / ar² = (1/9) / 9

r⁴ = 1/81

r⁴ = (1/3)⁴

So r = ±1/3.

Checking the options: Only 1/3 appears among the options (the negative value -1/3 is also mathematically valid, since (-1/3)⁴ = 1/81 as well, but it isn't listed).

Verification with r = 1/3:

• ar² = 9 gives a = 81
• T₇ = 81 × (1/3)⁶ = 81/729 = 1/9 ✓

Answer: A (1/3)

Tip: When you divide two terms of a G.P., the first term a cancels, leaving r raised to the difference in positions (here 7 − 3 = 4). Always remember that an even power gives both a positive and a negative root, so check which one the options include.