Question

1,2,4,8,16,32,64,______256

• A. 154
• B. 164
• C. 128
• D. 121

Hint:

Powers of 2 are fundamental in logical reasoning. Memorizing powers of 2 up to $2^{10}$ ($1024$) helps solve progression and binary-based series questions rapidly.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem requires identifying the underlying pattern of a geometric sequence.
In a geometric sequence, each successive term is obtained by multiplying the preceding term by a constant, non-zero number known as the common ratio ($r$).
Key Formula or Approach:
The general term of a geometric sequence is given by: \[ a_n = a_1 \cdot r^{n-1} \] Let's find the ratio between successive terms to identify $r$: \[ r = \frac{a_{i+1}}{a_i} \]

Step 2: Detailed Explanation:
Let's examine the terms given in the sequence: - First term ($a_1$) = $1 = 2^0$
- Second term ($a_2$) = $2 = 2^1$
- Third term ($a_3$) = $4 = 2^2$
- Fourth term ($a_4$) = $8 = 2^3$
- Fifth term ($a_5$) = $16 = 2^4$
- Sixth term ($a_6$) = $32 = 2^5$
- Seventh term ($a_7$) = $64 = 2^6$
The pattern reveals that the $n$-th term is represented as powers of 2: \[ a_n = 2^{n-1} \] To find the missing eighth term ($a_8$): \[ a_8 = 2^{8-1} = 2^7 \] \[ 2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128 \] Let's check if the pattern holds true for the final term given in the series, which is the ninth term ($a_9$): \[ a_9 = 2^8 = 256 \] The calculated term of 128 fits the sequence.

Step 3: Final Answer:
The missing number in the series is 128.