Can you use this to find the 10th, 17th and 80th triangular numbers?

Question 21 of 51beginner🔧 ApplyNumerical2 marks

Can you use this to find the $10^{th}$ , $17^{th}$ and $80^{th}$ triangular numbers?

Correct Answer

Yes, we can find the triangular numbers using the formula $t_n = \frac{n(n+1)}{2}$ .

Step 1: For the $10^{th}$ triangular number, $n = 10$ .

t_{10} = \frac{10 \times (10+1)}{2} = \frac{110}{2} = 55

Step 2: For the $17^{th}$ triangular number, $n = 17$ .

t_{17} = \frac{17 \times (17+1)}{2} = \frac{306}{2} = 153

Step 3: For the $80^{th}$ triangular number, $n = 80$ .

t_{80} = \frac{80 \times (80+1)}{2} = \frac{6480}{2} = 3240

Exercise: Think and Reflect | Q: 1 | (Page 12)

For More Understanding

Explanation

The textbook context establishes that the nth term of the sequence of triangular numbers is the sum of the first n natural numbers, which is given by the explicit rule $t_n = \frac{n(n+1)}{2}$ . By substituting the required term numbers (10, 17, and 80) into this formula, we can directly calculate their corresponding triangular numbers without having to sum the natural numbers sequentially.

Solution Steps

Step 1: State the formula for the nth triangular number: $t_n = \frac{n(n+1)}{2}$
Step 2: Calculate the 10th triangular number: $t_{10} = \frac{10 \times 11}{2} = 55$
Step 3: Calculate the 17th triangular number: $t_{17} = \frac{17 \times 18}{2} = 153$
Step 4: Calculate the 80th triangular number: $t_{80} = \frac{80 \times 81}{2} = 3240$

Previous Next