Question:medium

In the number series 4, 10, 23, 50, 104, 216, 439 the wrong number is

Show Hint

Try doubling and adding a growing number: 4 times 2 plus 2 gives 10, 10 times 2 plus 3 gives 23, 23 times 2 plus 4 gives 50. Carry on and see which printed term does not match.
Updated On: Jul 17, 2026
  • 10
  • 23
  • 104
  • 50
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Write out the gaps between terms.
Subtract each term from the next one:
$10 - 4 = 6$
$23 - 10 = 13$
$50 - 23 = 27$
$104 - 50 = 54$
$216 - 104 = 112$
$439 - 216 = 223$
So the gap sequence is 6, 13, 27, 54, 112, 223.

Step 2: Study the gaps.
Each gap is close to double the one before it. Checking exactly: $6 \times 2 + 1 = 13$, $13 \times 2 + 1 = 27$, but $27 \times 2 + 1 = 55$, not 54. Then $54 \times 2 + 1 = 109$, not 112, and $112 \times 2 + 1 = 225$, not 223.
The gap chain stays clean up to 27 and then goes wobbly right after, which points the finger at the term that follows 50.

Step 3: Repair the suspect term and re-test.
Replace 104 with 105. The gaps become 6, 13, 27, 55, 111, 223. Now check:
$6 \times 2 + 1 = 13$
$13 \times 2 + 1 = 27$
$27 \times 2 + 1 = 55$
$55 \times 2 + 1 = 111$
$111 \times 2 + 1 = 223$
Every single link now holds. One change repaired the whole chain, which is exactly what a single wrong number should do.

Step 4: Cross-check with the direct term rule.
The corrected series 4, 10, 23, 50, 105, 216, 439 obeys the rule $a_{n+1} = 2a_n + k$ where $k$ runs 2, 3, 4, 5, 6, 7:
$2(4)+2 = 10$, $2(10)+3 = 23$, $2(23)+4 = 50$, $2(50)+5 = 105$, $2(105)+6 = 216$, $2(216)+7 = 439$.
All six links agree, confirming that 105 belongs in the fifth slot.

Step 5: Rule out the rest.
Changing 10, 23 or 50 instead cannot fix the series, because each of those already sits correctly in both the term rule and the gap chain. Removing any of them would break links that currently work and would still leave the 104 problem untouched.

Final Answer:
The misfit is 104, which ought to be 105.
\[ \boxed{104} \]
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