Patterns #4

Arithmetic sequences

Definition

A sequence \(a_1, a_2, a_3, a_4, a_5,\ldots\) is called arithmetic if there is a number $d$ such that each term in the sequence can found by adding \(d\) to the previous term: \(a_1 \overset{+d}{\rightarrow} a_2 \overset{+d}{\rightarrow} a_3 \overset{+d}{\rightarrow} a_4 \overset{+d}{\rightarrow} a_5 \overset{+d}{\rightarrow}\cdots\). This means that \(a_n = a_{n-1} + d\) for each index $n\ge 2$. The number $d$ is called the common difference.

Exercise

Determine if each sequence is arithmetic or not. If it is arithmetic, determine the common difference.

  1.   \(3,5,7,11,13,17\ldots\)

  2.   \(2,8,14,20,26,32,\ldots\)

  3.   \(10,3,-4,-11,-18,-25,\ldots\)

  4.   \(2,4,8,16,32,64,\ldots\)

  5.   \(7, 9.25, 11.5, 13.75, 16, 18.25,\ldots\)

  6. The number of blocks in the sequence of figures beginning:

Figure 1
Figure 2
Figure 3
Figure 4


Exercise

Suppose that \(a_1, a_2, a_3, a_4, a_5,\ldots\) is an arithmetic sequence with common difference 4. Suppose that you also know \(a_2 = 23\). Determine each of \(a_1\) and \(a_3\), and then determine \(a_{100}\).





Arithmetic sequence formula

If \(a_1, a_2, a_3, a_4, a_5,\ldots\) is an arithmetic sequence with common difference \(d\), then a formula for the $n$th term is: \(a_n = a_1 +\)      

Exercise

Suppose that \(a_1, a_2, a_3, a_4, a_5,\ldots\) is an arithmetic sequence. If \(a_1 = 7\) and \(a_{33} = 359\), what is the common difference? Try using your formula above.


Exercise

An estimated 100 billion single-use plastic bags are used (and then thrown away) in the US each year. Each shopper uses on average 450 of these bags each year, and if 1% of shoppers stopped using these bags, then about 0.945 billion bags could be subtracted from the 100 billion that end up in the trash each year. Let’s study what would happen if an additional 1% of shoppers stopped using these bags each year. Let \(B_1\) be the number of bags in billions that are being used at the beginning of the 1st year of our study, so \(B_1 = 100\). Then \(B_2\) is the number of bags being used after the first 1% reduction, so \(B_2 = 100 - 0.945 = 99.055\).

  1. Determine \(B_3\), the number of bags being used at the start of the 3rd year (after two 1% reductions).

  2. Write out an explicit formula for $B_n$, and use it to compute \(B_{20}\).


  3. According to this model, at the beginning of what year would there be no more single-use plastic bags being thrown away?