The sum of five consecutive odd numbers is $575$. What is the sum of the next set of five consecutive odd numbers?

### Concept & Strategy When shifting from one set of consecutive numbers to the very next consecutive set, the total sum increases by a predictable fixed amount based on the number of items and the interval. ### Step-by-Step Solution * **Given:** The sum of five consecutive odd numbers is $575$. Let's find the numbers to visualize it. * The average (middle term) is $\frac{575}{5} = 115$. * The first set of odd numbers is: $111, 113, \textbf{115}, 117, 119$. * The "next set of five consecutive odd numbers" starts exactly after $119$. * The second set is: $121, 123, 125, 127, 129$. * Calculate the sum of this new set. The middle term is $125$. $$\text{Sum} = 125 \times 5 = 625$$ * Check the options: (a) $595$, (b) $615$, (c) $635$. The value $625$ is not listed. ### Exam Strategy & Shortcut You don't need to find the actual numbers. Compare the two sets term by term: Set 1: $A, B, C, D, E$ Set 2: $F, G, H, I, J$ (where $F$ is the odd number immediately following $E$) Since consecutive odd numbers have a gap of $2$, there are $5$ steps (a gap of $10$) between any term in the first set and the corresponding term in the second set (e.g., $F = A + 10$, $G = B + 10$). Because there are $5$ terms, the total sum increases by: $5 \times 10 = 50$. $$\text{New Sum} = 575 + 50 = 625$$ Since $625$ is absent, immediately mark "None of these". ### Common Pitfall A very common mistake is confusing "odd numbers" with regular "natural numbers." If they were natural numbers, the shift for $5$ items would be $+25$, resulting in $600$. Because they are odd numbers, the step size is $2$, doubling the aggregate shift to $+50$. ### Final Answer **Therefore, the correct answer is None of these.**