Finding Terms in a Sequence
A sequence is an ordered list of numbers. Sometimes, we need to determine the value of a specific term in a sequence.
One approach is to extend the sequence until we reach the desired term. Another approach is to find the general rule for the sequence and then evaluate for the term we need. While it is often easy to find the fifth or sixth term in a sequence by extending the pattern, this strategy gets more tedious the further the term of interest is from the known terms in the sequence.
Extending the Sequence
If we can determine the pattern in a sequence of numbers, we can continue the pattern until we reach the number we're looking for. For example, when given the pattern \(\frac{1}{3}, 1, 3, 9,...,\) we can see that we multiply each term by 3 to get the next term. Therefore, the value of the \(5^{\text{th}}\) term is \(9\times 3 = 27\) and the value of the \(6^{\text{th}}\) term is \(27 \times 3 = 81.\)
Find the \( 8^{\text{th}}\) term in the sequence \( 1, 4, 9, 16, 25, \ldots . \)
One way to complete this problem is to look for a pattern that allows us to find the next term and then follow the pattern until it reaches the term we are interested in. Note that the differences between the terms are increasing odd numbers \(( +3, +5, +7 \), etc.\(),\) and then continue in that manner until the \( 8^{\text{th}}\) term is reached. So, the \( 6^{\text{th}}\) term is \( 25 + 11 = 36 \), the \( 7^{\text{th}}\) is \(25 + 11 + 13 = 49, \) and the \( 8^{\text{th}}\) term is \( 25 + 11 + 13 + 15 = 64.\ _\square \)
Find the \( 7^{\text{th}}\) term in the sequence \( 2, 4, 8, 16, 32, \dots .\)
The pattern in this number sequence is to multiply a term by 2 to get the next term. The \(5^{\text{th}}\) term has a value of 32, so the \(7^{\text{th}}\) term has a value of \(32 \times 2 \times 2 = 128.\ _\square\)
Finding General Rule
Extending a sequence to find a specific value may not always be a realistic approach. For example, we wouldn't want to extend a sequence from the beginning to find the value of the \(200^{\text{th}}\) term.
Writing a general rule for a sequence provides a more elegant and oftentimes more efficient way to find unknown terms in a sequence.
Find the \( 12^{\text{th}}\) term in the sequence \( 1, 4, 9, 16, 25, \ldots . \)
We want to find a formula that expresses the \( n^\text{th} \) term directly as a function of \( n\). In this case, we can see that the first term is \( 1 \times 1 = 1 \), the second term is \( 2 \times 2 = 4\), and, in general, the \(n^\text{th}\) term is \( n^2 \). Thus the \( 12^{\text{th}}\) term is \( 12 \times 12 = 144.\ _\square\)
Find the \( 100^{\text{th}}\) term in the sequence \( 1, 3, 5, 7, 9, \dots .\)
Each term is increased by 2. Since the initial term is 1, the general term is\[a_n= 1 + 2(n-1) = 2n-1.\]
Thus, the \( 100^{\text{th}}\) term is \(2(100)-1 = 199.\ _\square\)
Find the \( 10^{\text{th}}\) term in the sequence \( 27, 9, 3, 1, \frac{1}{3}, \dots .\)
Observe that, given the initial term 27, the general term is \(a_n= 27 \cdot \left( \frac{1}{3}\right)^{n-1} .\) Then the \( 10^{\text{th}}\) term is\[27 \cdot \left( \frac{1}{3}\right)^{9} = \frac{3^3}{3^{9}}=\frac{1}{3^6}.\ _\square\]