Question

Class Activity 1B: Problems About Triangular Num-bersThe number of dots in a triangular array is called a triangular number.1st = 12nd = 33rd = 64th = 10The first triangular number is 1, the second is 3, the third is 6, the fourthis 10, and so on. In general, the Nth triangular number is the number ofdots in a triangular array of dots that has N dots on each side.1. What is the 25th triangular number? What is the 1000th triangu-lar number?

Answer 1

We have to find a sequence that describes the number of these triangular arrays , so we can determine the number of dots for the array number 25 and for the array number 1000.

Westart by noticing that array 1 (A1 ) = 1, The next one is created by adding 2 to it:

A1 = 1

A2 = 1 + 2

A3 = A2 + 3

A4 = A3 + 4

so every new step we add the "index" of the array to the "previous" array number (this is a recursive defiition).

An =A(n-1) + n

So let's observe how the sequence is created numerically:

A1 = 1

A2 = 1 + 2

A3 = 1 + 2 + 3

A4 = 1 + 2 + 3 + 4

A5 = 1 + 2 + 3 + 4 + 5

A6 = 1 + 2 + 3 + 4 + 5 + 6

WHich reminds us of the arithmetic sequence , given that each term is the arithmetic sequence starting at 1 and ending at the number of the ekement in question.

We can therefore used Gauss' formula for that:

Partial sum = n (A1 + An) /2

Thenfore the term 25 we have:

25 (1 + 25) / 2 = 25 * 26 / 2 = 325

and for the term 1000 we have:

1000 (1 + 1000) / 2 = 500500