Sequences are ordered lists of numbers that each follows a specific pattern. These patterns allow you to find the equation either explicitly or recursively. The pattern may be arithmetic, geometric, or neither.
Arithmetic Sequences come from consecutive terms having a constant difference, \(d\), between them.
Geometric Sequences come from consecutive terms having a constant ratio, \(r\), between them
Here are some important definitions to consider when looking at set notation for sequences:
A sequence can either be finite (meaning it has a concrete final term) or infinite (meaning it has no end). Gaps between terms can be displayed using ellipses. If no term comes after the ellipses, that indicates the sequence is infinite.
The respective formulas for finding the common differences and ratios of different sequences can be algebraically expressed as such:
When identifying whether a sequence is either arithmetic or geometric, make sure to use AT LEAST \(2\) sets of values to prove its identity!
When writing an explicit equation, you write it in terms of its first term, \(a\), and position, \(n\). When writing a recursive equation, you write it in terms of previous terms, \(t_{n-1}\).
The general explicit formulas for Arithmetic and Geometric Sequences can be respectively expressed as such:
The recursive formulas for Arithmetic and Geometric Sequences can be respectively expressed as such:
NOTE: The first term must be listed after the recursive formula since there is no previous term(s) it relies on to determine its value.
For the sequence \(-4,-7,-10,-13,...,-52\), identify the following:
i. We can determine that this sequence is artithmetic. To prove this, we can use the difference formula to prove that there is a constant difference between terms:
We can first determine the difference between the second and first terms:
\(\textcolor{green}{\textcolor{green}{d_1}} = (\textcolor{teal}{-7}) - (\textcolor{brown}{-4})\)
\(\textcolor{green}{d_1} = \textcolor{green}{-3}\)
We can then determine the difference between the third and second terms:
\(\textcolor{green}{d_2} = (\textcolor{teal}{-10}) - (\textcolor{brown}{-7})\)
\(\textcolor{green}{d_2} = \textcolor{green}{-3}\)
Finally, we can compare the two differences to determine that they're the same:
\(\textcolor{green}{d_1} = \textcolor{green}{d_2}\)
\(\textcolor{green}{-3} = \textcolor{green}{-3}\)
ii. We can determine that this sequence is finite since it has an ending term (in this case, \(-52\)).
iii. We can determine the explicit formula for tₙ using the first term of the sequence (\(-4\)) and the difference (\(-3\)):
\(\textcolor{teal}{t_n}= \textcolor{red}{a} + \textcolor{green}{d}(\textcolor{blue}{n}-1)\)
\(\textcolor{teal}{t_n}= \textcolor{red}{-4} \textcolor{green}{-3}(\textcolor{blue}{n}-1)\)
\(\textcolor{teal}{t_n}= -4 -3\textcolor{blue}{n} + 3\)
\(\textcolor{teal}{t_n}= -3\textcolor{blue}{n} -1\)
iv. Using the general explicit formula we identified in the last step, we can determine the next term of the sequence:
\(\textcolor{teal}{t_5}= -3(\textcolor{blue}{5}) -1\)
\(\textcolor{teal}{t_5}= -15 -1\)
\(\textcolor{teal}{t_5}= -16\)
v. The recursive formula can be determined using the previous term in the sequence, \(t_{n-1}\), and the difference, \(d\):
\(\textcolor{teal}{t_n}= (\textcolor{brown}{t_{n-1}})+\textcolor{green}{d}, t₁ =\textcolor{red}{a}\)
\(\textcolor{teal}{tₙ}= (\textcolor{brown}{t_{n-1}})\textcolor{green}{-3}, t₁ =\textcolor{red}{-4}\)
For the sequence \(5x,7x,9x,...\), identify the following:
i. We can determine that this sequence is artithmetic. To prove this, we can use the difference formula to prove that there is a constant difference between terms.
We can first determine the difference between the second and first terms:
\(\textcolor{green}{d_1} = \textcolor{teal}{7x} - \textcolor{brown}{5x}\)
\(\textcolor{green}{d_1} = 2x\)
We can then determine the difference between the third and second terms:
\(\textcolor{green}{d_2} = \textcolor{teal}{9x} - \textcolor{brown}{7x}\)
\(\textcolor{green}{d_2} = 2x\)
Finally, we can compare the two differences to determine that they're the same:
\(\textcolor{green}{d_1} = \textcolor{green}{d_2}\)
\(\textcolor{green}{2x} = \textcolor{green}{2x}\)
ii. We can determine that this sequence is infinite since it has no ending term.
iii. We can determine the explicit formula for \(t_n\) using the first term of the sequence (\(5x\)) and the difference (\(2x\)):
\(\textcolor{teal}{t_n}= \textcolor{red}{a} + \textcolor{green}{d}(\textcolor{blue}{n}-1)\)
\(\textcolor{teal}{t_n}= 5x +2x(\textcolor{blue}{n}-1)\)
\(\textcolor{teal}{t_n}= 5x + 2x\textcolor{blue}{n} -2x\)
\(\textcolor{teal}{t_n}= 2x(\textcolor{blue}{n}) + 3x\)
iv. Using the general explicit formula we identified in the last step, we can determine the next term of the sequence:
\(\textcolor{teal}{t_4}= 2x(\textcolor{blue}{4}) + 3x\)
\(\textcolor{teal}{t_4}= 8x + 3x\)
\(\textcolor{teal}{t_4}= 11x\)
v. The recursive formula can be determined using the previous term in the sequence, \(t_{n-1}\), and the difference, \(d\):
\(\textcolor{teal}{t_n}= (\textcolor{brown}{t_{n-1}})+\textcolor{green}{d}, t₁ =\textcolor{red}{a}\)
\(\textcolor{teal}{t_n}= (\textcolor{brown}{t_{n-1}})+\textcolor{green}{2x}, t₁ =\textcolor{red}{5x}\)
For the sequence \(32+16+8+4+...\), identify the following:
i. We can determine that this sequence is geometric. To prove this, we can use the ratio formula to prove that there is a constant ratio between terms.
We can first determine the ratio between the second and first terms:
We can then determine the ratio between the third and second terms:
Finally, we can compare the two ratios to confirm that they're the same:
\(\textcolor{magenta}{r_1} = \textcolor{magenta}{r_2}\)
\(\textcolor{magenta}{\cfrac{1}{2}} = \textcolor{magenta}{\cfrac{1}{2}}\)
ii. We can determine that this sequence is infinite since it has no ending term.
iii. We can determine the explicit formula for \(t_n\) using the first term of the sequence (\(32\)) and the ratio \(\left(\cfrac{1}{2}\right)\):
\(\textcolor{teal}{t_n}= \textcolor{red}{a}(\textcolor{green}{r})^{\textcolor{blue}{n}-1}\)
\(\textcolor{teal}{t_n}= \textcolor{red}{32}\left(\textcolor{green}{\cfrac{1}{2}}\right)^{\textcolor{blue}{n}-1}\)
iv. Using the general explicit formula we identified in the last step, we can determine the next term of the sequence:
\(\textcolor{teal}{t_5}= \textcolor{red}{32}\left(\textcolor{green}{\cfrac{1}{2}}\right)^{\textcolor{blue}{5}-1}\)
\(\textcolor{teal}{t_5}= \textcolor{red}{32}\left(\cfrac{1}{2}\right)^4\)
\(\textcolor{teal}{t_5}= \textcolor{red}{32}\left(\cfrac{1}{16}\right)\)
\(t_5= 2\)
v. The recursive formula can be determined using the previous term in the sequence, \(t_{n-1}\), and the ratio, \(r\):
\(\textcolor{teal}{t_n}= (\textcolor{brown}{t_{n-1}}(\textcolor{magenta}{r}), t_1 =\textcolor{red}{a}\)
\(\textcolor{teal}{t_n}= (\textcolor{brown}{t_{n-1}})\left(\cfrac{\textcolor{magenta}{1}}{\textcolor{magenta}{2}}\right), t_1 =\textcolor{red}{32}\)
For the sequence \(\sqrt{3},\sqrt{6},\sqrt{9},\sqrt{12}...\sqrt{33}\), identify the following:
i. We can determine that this sequence is neither arithmetic or geometric. To prove this, we can use the difference and ratio formula to prove that there's no constant pattern between terms.
We will first determine the differences to identify whether they are the same using the following formula:
We can first determine the difference between the second and first terms:
\(\textcolor{green}{d_1} = \textcolor{teal}{\sqrt{6}} - \textcolor{brown}{\sqrt{3}}\)
\(\textcolor{green}{d_1} = 2.45 - 1.73\)
\(\textcolor{green}{d_1} = 0.72\)
We can then determine the difference between the third and second terms:
\(\textcolor{green}{d_2} = \sqrt{\textcolor{teal}{9}} - \sqrt{\textcolor{brown}{6}}\)
\(\textcolor{green}{d_2} = 3 - 2.45\)
\(\textcolor{green}{d_2} = 0.55\)
We can now compare the two differences to determine if they're the same:
\(\textcolor{green}{d_1} \ne \textcolor{green}{d_2}\)
\(\textcolor{green}{0.72} \ne \textcolor{green}{0.55}\)
Since the differences are different, we will now determine the ratios to identify whether they are the same using the following formula:
We can first determine the ratio between the second and first terms:
\(\textcolor{magenta}{r_1} = \cfrac{\textcolor{teal}{\sqrt{6}}}{\textcolor{brown}{\sqrt{3}}} = \cfrac{2.45}{1.73} = 1.42\)
We can then determine the ratio between the third and second terms:
\(\textcolor{magenta}{r_2} = \cfrac{\textcolor{teal}{\sqrt{9}}}{\textcolor{brown}{\sqrt{6}}} = \cfrac{3}{2.45} = 1.22\)
We can now compare the two ratios to determine if they're the same:
\(\textcolor{magenta}{r_1} \ne \textcolor{magenta}{r₂}\)
\(\textcolor{magenta}{1.42} \ne \textcolor{magenta}{1.22}\)
As indicated above, neither pair of differences or ratios are equal to each other. This confirms that this sequence is neither arithmetic or geometric.
ii. We can determine that this sequence is finite since it has an ending term (in this case \(\sqrt{33}\)).
iii. We can determine the explicit formula for \(t_n\) by multiplying the first term in the sequence (\(\sqrt{3}\)) by the term number, \(n\):
iv. Using the general explicit formula we identified in the last step, we can determine the next term of the sequence:
\(\textcolor{teal}{t_5}= \sqrt{3(\textcolor{blue}{5})}\)
\(\textcolor{teal}{t_5}= \sqrt{15}\)
v. The recursive formula can be determined using the previous term in the sequence, \(t_{n-1}\), and the starting term, \(a\):
\(\textcolor{teal}{t_n}= (\textcolor{brown}{t_{n-1}} + \textcolor{red}{a}, t_1 =\textcolor{red}{a}\)
\(\textcolor{teal}{t_n}= \sqrt{((\textcolor{brown}{t_{n-1}})^2+\textcolor{red}{3})}, t₁ =\textcolor{red}{\sqrt{3}}\)