Question

When we combine two consecutive triangular numbers, we get a______
a. Square number
b. Consecutive number
c. Non square number
d. Zero

Answer 1

Hint: From the question we have been asked to find the result we get when we combine two consecutive triangular numbers. so, for solving this question we will take the help of definition of the triangular numbers. After using the definition of triangular numbers we will simplify the solution using an example. So, we proceed with our solution as follows.

Complete step by step solution:
Firstly, the meaning of the triangular numbers is as follows. The triangular numbers are obtained by continued summation of the natural numbers.
So, to get the triangular numbers first we will take the natural number and add $2$ to it. So, we get the next number as $3$.
Now we will add $3$ to $3$. So, we get the next triangular number as $6$. We will continue this process and find the next triangular number.
So, the triangular numbers will be as follows.
$\Rightarrow 1,3,6,10,15....$
So, after getting these numbers we will add the consecutive triangular numbers and check the similarity in between them and find the solution to our question.
So, we take the first two consecutive triangular numbers which are $6$ and $3$. Now we will add these both. So, we get,
$\Rightarrow 1+3$
$\Rightarrow 4$
Now this can also be written as square of number two.
$\Rightarrow 4={{2}^{2}}$
Now we will take another example that is we will add the next two consecutive triangular numbers. So, we get,
$\Rightarrow 6+10$
$\Rightarrow 16$
We can also write this as square of number four.
$\Rightarrow 16={{4}^{2}}$
Therefore, we get the solution as a square number.

Note: Students must have good knowledge in the concept of triangular numbers to solve this type of questions. Students must not make mistakes like they must not add the numbers $1$ and $6$ as they are not consecutive if they do then our solution will be wrong.