What are the two consecutive natural numbers whose sum of squares is equal to 145?

Introduction / Context:
This question asks you to find two consecutive natural numbers whose squares add up to 145. It tests your ability to model consecutive numbers with algebra, set up an equation using their squares and then solve the resulting quadratic equation. Consecutive number problems appear frequently in aptitude tests and help build comfort with simple algebraic modeling.

Given Data / Assumptions:

- There are two consecutive natural numbers. - Let the smaller one be n, so the next is n + 1. - Their sum of squares is 145. - We are looking for natural number solutions, that is positive integers.

Concept / Approach:
Consecutive numbers can be easily expressed as n and n + 1. The condition on the sum of squares translates into the equation n^2 + (n + 1)^2 = 145. Expanding and simplifying this expression will give a quadratic equation in n. Solving this equation yields the possible values of n, and we then check which solutions are valid natural numbers. Finally, we report the pair that satisfies the original condition.

Step-by-Step Solution:
Step 1: Let the two consecutive natural numbers be n and n + 1. Step 2: Write the condition on their squares: n^2 + (n + 1)^2 = 145. Step 3: Expand (n + 1)^2: (n + 1)^2 = n^2 + 2n + 1. Step 4: Substitute into the equation: n^2 + n^2 + 2n + 1 = 145. Step 5: Combine like terms: 2n^2 + 2n + 1 = 145. Step 6: Subtract 145 from both sides: 2n^2 + 2n + 1 - 145 = 0, so 2n^2 + 2n - 144 = 0. Step 7: Divide the entire equation by 2 to simplify: n^2 + n - 72 = 0. Step 8: Factor the quadratic: n^2 + n - 72 = (n + 9)(n - 8) = 0. Step 9: Set each factor equal to zero: n + 9 = 0 or n - 8 = 0. Step 10: So n = -9 or n = 8. Step 11: Since we need natural numbers, we discard n = -9 and keep n = 8. Step 12: The consecutive numbers are therefore 8 and 9.

Verification / Alternative check:
Verify by direct substitution. Compute 8^2 + 9^2 = 64 + 81 = 145, which matches the required sum. None of the other option pairs have this property. For example, 6^2 + 7^2 = 36 + 49 = 85, and 9^2 + 10^2 = 81 + 100 = 181, both of which differ from 145. This confirms that 8 and 9 are the correct numbers.

Why Other Options Are Wrong:
The pairs 6 and 7, 13 and 14, and 9 and 10 do not satisfy the sum of squares condition. Their squared sums are 85, 365 and 181 respectively, none of which equal 145. These values may come from miscalculating the quadratic or guessing without checking. Only the pair 8 and 9 produces the correct total of 145 for the sum of squares.

Common Pitfalls:
Some learners may misinterpret the phrase "consecutive natural numbers" and treat the numbers as n and n + 2 or use only one equation and guess the second number. Others might miscalculate the square or forget to combine all terms correctly when simplifying the quadratic. Taking care with the algebraic expansion and factoring ensures a smooth path to the correct solution.

Final Answer:
The required consecutive natural numbers are 8 and 9, which matches option A.