Question

Let $t_1, t_2, t_3, \ldots, t_{2n}$ be in G.P. with common ratio $r$. Then:

• A. $t_1,t_3,t_5,\ldots,t_{2n-1}$ are in G.P. with common ratio $r$
• B. $t_1,t_4,t_7,\ldots,t_{2n-1}$ are in G.P. with common ratio $r^2$
• C. $t_1,t_3,t_5,\ldots,t_{2n-1}$ are in G.P. with common ratio $r^2$
• D. $t_2,t_4,t_6,\ldots,t_{2n}$ are in G.P. with common ratio $r^3$
• E. $t_2,t_4,t_6,\ldots,t_{2n}$ are in G.P. with common ratio $r^5$

Hint:

Skipping one term in G.P. multiplies power of $r$ by 2.

Answer 1

The Correct Option is C

Concept:
• In G.P., $t_n = ar^{n-1}$
• Selecting alternate terms forms a new G.P.

Step 1: Write general terms
\[ t_1 = a,\quad t_2 = ar,\quad t_3 = ar^2,\quad t_4 = ar^3,\ldots \]

Step 2: Take odd-indexed terms
\[ t_1, t_3, t_5, \ldots = a, ar^2, ar^4, \ldots \]

Step 3: Find common ratio
\[ \frac{ar^2}{a} = r^2,\quad \frac{ar^4}{ar^2} = r^2 \] Thus, they form a G.P. with common ratio $r^2$ Final Conclusion:
Option (C) is correct.