In the series 5, 11, 21, 44, ?, 175, find the missing number by analysing the pattern that doubles each term and then adjusts by a small integer.

Introduction / Context:
This number series 5, 11, 21, 44, ?, 175 exhibits significant growth from term to term. The pattern appears to involve doubling each term and then adding or subtracting a small, systematically changing integer. Recognising this structure allows us to determine the missing value in the series.

Given Data / Assumptions:
The known terms are:

• 5
• 11
• 21
• 44
• ?
• 175

We assume there is a fixed two step rule that takes each term to the next by using multiplication by 2 and then adding or subtracting a small sequence of integers that alternates in sign and increases.

Concept / Approach:
A useful idea is to test whether the rule is of the form a(n+1) = 2 * a(n) ± k, where the sign and the value of k follow a simple pattern such as +1, −1, +2, −2, +3, etc. We check this for consecutive transitions and see whether it reproduces the given terms and leads to a unique missing value that matches the last term 175.

Step-by-Step Solution:
Step 1: From 5 to 11. Check 5 * 2 + 1 = 10 + 1 = 11. So here we double and add 1.Step 2: From 11 to 21. Check 11 * 2 − 1 = 22 − 1 = 21. So we now double and subtract 1.Step 3: From 21 to 44. Check 21 * 2 + 2 = 42 + 2 = 44. Now we double and add 2.Step 4: This suggests a pattern: multiply by 2 and then alternately add and subtract, with the magnitude of the integer increasing: +1, −1, +2, −2, +3, ... .Step 5: From 44 to the missing term, we should apply the next stage of the pattern, which is double then subtract 2: 44 * 2 − 2 = 88 − 2 = 86.Step 6: From 86 to 175, check the final step: 86 * 2 + 3 = 172 + 3 = 175. This follows the next part of the pattern, double then add 3.

Verification / Alternative check:
We can summarise the sequence with the rule a(n+1) = 2 * a(n) + s, where s follows the cycle +1, −1, +2, −2, +3. Applying it from the start gives 5 → 11 → 21 → 44 → 86 → 175, perfectly matching all given terms when the missing term is 86. Because the final known term 175 also fits the rule when 86 is used, this strongly confirms the consistency of our solution.

Why Other Options Are Wrong:
Values 78, 94 and 102 would not allow the transform 2 * a(n) + s to generate 175 using the next adjustment of +3, and they would break the simple alternating pattern of ±1, ±2, ±3 in the earlier steps. Only 86 satisfies both the intermediate step from 44 and the final step to 175 with a neat and consistent adjustment sequence.

Common Pitfalls:
Many students look only at the raw differences between terms (6, 10, 23, ?) and do not notice the more structured rule involving doubling plus or minus small integers. When numbers grow roughly by a factor of 2, always test rules of the form 2 * a(n) ± k rather than relying solely on first differences.

Final Answer:
The missing number that keeps the doubling and alternating adjustment pattern consistent is 86.