Triangular Numbers

Triangular numbers are a sequence of numbers that are formed by adding consecutive positive integers. They are called triangular because they can be represented visually as dots arranged in an equilateral triangle.

The first few triangular numbers are:

1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21
1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36

Formula for Triangular Numbers

The formula for the nth triangular number is:

T_n = \frac{n(n+1)}{2}

Here, T_n represents the nth triangular number, and n is the position of the triangular number in the sequence.

Examples Using the Formula

1. Find the 8th triangular number.

Using the formula with n=8n=8:

T_8 = \frac{8(8+1)}{2} = \frac{8 \times 9}{2} = 4 \times 9 = 36T_8 = \frac{8(8+1)}{2} = \frac{8 \times 9}{2} = 4 \times 9 = 36

2. Find the 20th triangular number.

Using the formula with n=20n=20:

T_{20} = \frac{20(20+1)}{2} = \frac{20 \times 21}{2} = 10 \times 21 = 210T_{20} = \frac{20(20+1)}{2} = \frac{20 \times 21}{2} = 10 \times 21 = 210

3. Find the 100th triangular number.

Using the formula with n=100n=100:

T_{100} = \frac{100(100+1)}{2} = \frac{100 \times 101}{2} = 50 \times 101 = 5050T_{100} = \frac{100(100+1)}{2} = \frac{100 \times 101}{2} = 50 \times 101 = 5050

Identifying Triangular Numbers

To determine if a given number is a triangular number, we can use the formula \frac{n(n+1)}{2} and solve for n.

4. Is 325 a triangular number?

We want to see if there is an integer value of nn for which \frac{n(n+1)}{2} = 325\frac{n(n+1)}{2} = 325.

n(n+1) = 325 \times 2n(n+1) = 325 \times 2

n(n+1) = 650n(n+1) = 650

By trial and error, we find that 25 \times 26 = 65025 \times 26 = 650, so n = 25n = 25. Therefore, 325 is indeed a triangular number.

You’ve used 0 of your 10 free revision notes for the month

Sign up to get unlimited access to revision notes, quizzes, audio lessons and more

Sign up