Question

Which of the following defines any term in a linear sequence having 30 for its first term and 126 for its ninth term?

• A. \( s_n = s_{n-1} + \frac{16}{3} \)
• B. \( s_n = s_{n-1} + 8 \)
• C. \( s_n = s_{n-1} + 12 \)
• D. \( s_n = 2s_{n-1} + 4 \)
• E. \( s_n = s_{n-1} + \frac{32}{3} \)

Hint:

Always use the \( a_n = a_1 + (n-1)d \) formula to check consistency of linear (arithmetic) sequences.

Answer 1

The Correct Option is C

Step 1: Formula for nth term.
For an arithmetic sequence: \[ a_n = a_1 + (n-1)d \] Here \( a_1 = 30 \), \( a_9 = 126 \).
Step 2: Use given values.
\[ 126 = 30 + (9-1)d \quad \Rightarrow \quad 126 = 30 + 8d \] \[ 96 = 8d \quad \Rightarrow \quad d = 12 \]
Step 3: Write recurrence relation.
So, \[ s_n = s_{n-1} + 12 \]
Final Answer: \[ \boxed{s_n = s_{n-1} + 12} \]