A, B, C, D and E are five consecutive odd numbers. The sum of A and C is $146$. What is the value of E?

### Concept & Logic Consecutive odd numbers always have a common difference of $2$. By defining the first term, the rest follow predictably. $$A = x, B = x+2, C = x+4, D = x+6, E = x+8$$ ### Step-by-Step Solution * **Given:** We have five consecutive odd numbers: $A$, $B$, $C$, $D$, and $E$. * **Given:** The sum of $A$ and $C$ is $146$. * Substitute the algebraic expressions for $A$ and $C$ into the equation: $$x + (x + 4) = 146$$ * Combine like terms: $$2x + 4 = 146$$ * Isolate the variable term: $$2x = 142$$ * Divide to find the value of the first odd number ($A$): $$x = 71$$ * We need to find the value of $E$, which is $x + 8$: $$E = 71 + 8 = 79$$ ### Exam Strategy & Shortcut Because consecutive odd numbers form an Arithmetic Progression (AP), the middle term between any two symmetric terms is their average. The number exactly between $A$ and $C$ is $B$. Therefore, $B$ is the average of $A$ and $C$: $B = \frac{146}{2} = 73$. Since the sequence is $A, B, C, D, E$ with a difference of $2$, you can just count up from $B$: $B = 73 \Rightarrow C = 75 \Rightarrow D = 77 \Rightarrow E = 79$. This skips the formal algebra entirely. ### Common Pitfall A frequent mistake is solving for $x$ (getting $71$) and immediately selecting option (a) without re-reading the final question. The problem asks for the value of $E$, not $A$. Always underline the final target variable in the question. ### Final Answer **Therefore, the correct answer is 79.**