Questions: Use inductive reasoning to answer. If you add two consecutive triangular numbers, what kind of figurate number do you get? Choose the correct answer below. A. a hexagonal number B. a square number C. a pentagonal number D. the next triangular number

Solution Steps

Triangular numbers are a sequence of numbers where each number represents a triangle with dots. The $n$-th triangular number $T_n$ is given by the formula: $$T_n = \frac{n(n+1)}{2}.$$

Let’s consider two consecutive triangular numbers, $T_n$ and $T_{n+1}$. Their sum is: $$T_n + T_{n+1} = \frac{n(n+1)}{2} + \frac{(n+1)(n+2)}{2}.$$

Combine the terms: $$T_n + T_{n+1} = \frac{n(n+1) + (n+1)(n+2)}{2}.$$ Factor out $(n+1)$: $$T_n + T_{n+1} = \frac{(n+1)(n + n + 2)}{2} = \frac{(n+1)(2n + 2)}{2}.$$ Simplify further: $$T_n + T_{n+1} = \frac{2(n+1)(n+1)}{2} = (n+1)^2.$$

The sum of two consecutive triangular numbers is $(n+1)^2$, which is a perfect square. Therefore, the result is a square number.

Final Answer