Product pattern n*(2n+1): find the odd one out Numbers: 3, 10, 21, 36, 55, 70, 105. Which number does not fit n*(2n+1)?

Introduction / Context:
Some sequences follow the form n*(2n+1) for consecutive integers n. Spotting this form helps check whether each number belongs. If exactly one number breaks the identity, that term is the odd one out.

Given Data / Assumptions:

• Candidates: 3, 10, 21, 36, 55, 70, 105.
• Proposed form: n*(2n+1) for integer n ≥ 1.

Concept / Approach:
Compute n*(2n+1) for n = 1, 2, 3, ... and compare with the terms. If one term cannot be expressed this way for any integer n while neighbors can, it is the odd one out.

Step-by-Step Solution:
n=1 ⇒ 1*(3)=3 ✔n=2 ⇒ 2*(5)=10 ✔n=3 ⇒ 3*(7)=21 ✔n=4 ⇒ 4*(9)=36 ✔n=5 ⇒ 5*(11)=55 ✔n=6 ⇒ 6*(13)=78 (not 70) ✘n=7 ⇒ 7*(15)=105 ✔Thus 70 does not fit the pattern; expected value between 55 and 105 would have been 78.

Verification / Alternative check:
Check differences: 3→10 (+7), 10→21 (+11), 21→36 (+15), 36→55 (+19), 55→78 (+23 expected), then 78→105 (+27). The +23 step is missing in the list and 70 appears instead, confirming the inconsistency.

Why Other Options Are Wrong:
105, 36, 55, and 21 each exactly match n*(2n+1) for n = 7, 4, 5, and 3 respectively. They fit the rule.

Common Pitfalls:
Assuming a constant difference or trying to fit a linear rule when the quadratic-like growth suggests a formula such as n*(2n+1).

Final Answer:
70