Question

For a sequence \( (t_n) \), if \( s_n = 7(3^n - 1) \), then \( t_n = \)

• A. \( 7 \cdot 3^{n-1} \)
• B. \( 14 \cdot 3^{n+1} \)
• C. \( 14 \cdot 3^{n-1} \)
• D. \( 7 \cdot 3^{n+1} \)

Hint:

Always remember: \( t_n = s_n - s_{n-1} \) when a sequence is given in terms of its partial sums.

Answer 1

The Correct Option is C

Step 1: Use the relation between \( s_n \) and \( t_n \).
We know that \[ t_n = s_n - s_{n-1} \] Step 2: Substitute the given expression.
\[ s_n = 7(3^n - 1), \quad s_{n-1} = 7(3^{n-1} - 1) \] Step 3: Find \( t_n \).
\[ t_n = 7(3^n - 1) - 7(3^{n-1} - 1) \] \[ = 7(3^n - 3^{n-1}) \] Step 4: Simplify.
\[ t_n = 7 \cdot 3^{n-1}(3 - 1) = 14 \cdot 3^{n-1} \] Step 5: Conclusion.
Hence, \( t_n = 14 \cdot 3^{n-1} \).