Problem of the Week

Problem B and Solution

In an Orderly Fashion

Problem

1. When we write the year 2021, we are writing two consecutive two-digit numbers (20 and 21). Find all the other years from 1000 to 2021 that are made up of two consecutive two-digit numbers written in order, and add them to the table below.

2. Find the sum of the consecutive two-digit numbers for each year from part a), and add this to the table below. For example, for 2021, the sum is $$20 + 21 = 41$$. Describe the pattern formed by these sums.

3. Find the product of the consecutive two-digit numbers for each of the first 5 years in the table. Then find the differences of these products, in order. For example, $$10\times 11=110$$ and $$11\times 12=132$$. The difference is $$132-110=22$$.
You will have five products and four differences.

1. What pattern is formed by the differences?

2. Use this pattern to find the remaining products, without multiplying.

4. What sequence of numbers can you form by combining the numbers in the sum column and the difference column?

Year Sum Product Difference
1011 $$10+11 = 21$$ $$10\times 11 = 110$$
1112 $$11+12 = 23$$ $$11\times 12 = 132$$ $$132-110 = 22$$

2021 $$20+21=41$$ $$20\times 21 = 420$$

Solution

1. There are 11 years from 1000 to 2021 that are made up of consecutive two-digit numbers written in order. They are

1011, 1112, 1213, 1314, 1415, 1516, 1617, 1718, 1819, 1920, and 2021

2. The table shows that each sum is 2 more than the previous sum.

1. Each difference is 2 more than the previous difference.
If you fill in the difference column, then you can use that to fill in the product column. For example, the difference of products for 1112 is 22. This means that the next difference will be 24. Adding $$132+24=156$$, which is the next product. The entire table can be completed this way.

2. The completed table is below.

Year Sum Product Difference
1011 21 110
1112 23 132 22
1213 25 156 24
1314 27 182 26
1415 29 210 28
1516 31 240 30
1617 33 272 32
1718 35 306 34
1819 37 342 36
1920 39 380 38
2021 41 420 40
3. If we combine the numbers in these 2 columns, we get the sequence of whole numbers from 21 to 41.