Question

Which one of the given options is a possible value of $x$ in the following sequence? \[ 3,\; 7,\; 15,\; x,\; 63,\; 127,\; 255 \]

• A. 35
• B. 40
• C. 45
• D. 31

Hint:

When solving number sequences, always check for powers of $2$ or $3$, factorials, or prime-related patterns. Recognizing $2^n - 1$ often indicates Mersenne numbers.

Answer 1

The Correct Option is D

Step 1: Observe the pattern of the sequence. The terms given are: \[ 3,\; 7,\; 15,\; x,\; 63,\; 127,\; 255 \] Notice: \[ 3 = 2^2 - 1,\quad 7 = 2^3 - 1,\quad 15 = 2^4 - 1 \] So, the terms appear to follow the form: \[ 2^n - 1 \] Step 2: Verify later terms. \[ 63 = 2^6 - 1, \quad 127 = 2^7 - 1, \quad 255 = 2^8 - 1 \] This confirms the sequence pattern: \[ 2^2 - 1,\; 2^3 - 1,\; 2^4 - 1,\; 2^5 - 1,\; 2^6 - 1,\; 2^7 - 1,\; 2^8 - 1 \] Step 3: Identify the missing term. The missing term corresponds to $2^5 - 1$: \[ 2^5 - 1 = 32 - 1 = 31 \] \[ \boxed{31} \]