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

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

Transcript text: 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

Solution Steps

Step 1: Understanding Triangular Numbers

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: \[ T_n = \frac{n(n+1)}{2}. \]

Step 2: Adding Two Consecutive Triangular Numbers

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

Step 3: Simplifying the Sum

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: \[ T_n + T_{n+1} = \frac{2(n+1)(n+1)}{2} = (n+1)^2. \]

Step 4: Interpreting the Result

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