Question

If a1 + a2 + a3 + .... + an = 3(2n+1 - 2 ) , for every n≥ 1, then a11 equals

• A. 6144
• B. 9144
• C. 6441
• D. 9741

Answer 1

The Correct Option is A

Given: The sequence satisfies the relation: 

\[ a_1 + a_2 + \ldots + a_n = 3(2^{n+1} - 2) \]

Step-by-step evaluation:

• For \( n = 1 \):
  \[ a_1 = 3(2^{1+1} - 2) = 3(4 - 2) = 6 = 3 \times 2^1 \]
• For \( n = 2 \):
  \[ a_1 + a_2 = 3(2^{2+1} - 2) = 3(8 - 2) = 18 \Rightarrow a_2 = 18 - a_1 = 18 - 6 = 12 = 3 \times 2^2 \]
• For \( n = 3 \):
  \[ a_1 + a_2 + a_3 = 3(2^{3+1} - 2) = 3(16 - 2) = 42 \Rightarrow a_3 = 42 - (6 + 12) = 24 = 3 \times 2^3 \]

General Term:

\[ a_n = 3 \times 2^n \]

To find: \( a_{11} \)

\[ a_{11} = 3 \times 2^{11} = 3 \times 2048 = \boxed{6144} \]