Question

Which is the missing number in the given number sequence?
7, 15, 32, -----, 138, 281

• A. 67
• B. 65
• C. 69
• D. 63

Hint:

To find the pattern of rapidly increasing series, look at the ratio of consecutive terms. Since \( 15/7 \approx 2 \), \( 32/15 \approx 2 \), and \( 281/138 \approx 2 \), the pattern is highly likely to involve multiplication by 2.

Answer 1

The Correct Option is A

Step 1: List the terms.
The sequence is $7,15,32,-,138,281$.
Step 2: Guess a multiply-then-add rule.
Try $T_n = 2\,T_{n-1} + (n-1)$.
Step 3: Check the early terms.
$2(7)+1=15$ and $2(15)+2=32$, both correct.
Step 4: Compute the missing term.
$T_4 = 2(32)+3 = 64+3 = 67$.
Step 5: Verify forward.
$2(67)+4=138$ and $2(138)+5=281$, both match the given values.
Step 6: State the answer.
The missing number is $67$.
\[ \boxed{67} \]