Suppose the odd numbers are grouped in the following way: What is the sum of the numbers in the tenth grouping?

Source: NCTM Mathematics Teacher 2006

**SOLUTION**

*Brute force method
*It doesn’t take long to write out the consecutive odd numbers and notice that the tenth group consists of . The sum of the numbers in the tenth grouping equals .

*Pattern method
*Group has elements

Value of first element

Value of last element

Sum of elements

Group has elements

Value of first element

Value of last element

Sum of elements

Group has elements

Value of first element

Value of last element

Sum of elements

In general group has elements

Value of first element

Value of last element

We prove that the sum of elements .

Recall that given consecutive odd numbers

Sum of elements of group

Sum of the numbers in the tenth grouping

**Answer**: