# Finding Specific Terms in a Pattern

There are two general approaches to looking for a specific term in a sequence: extending the sequence until it reaches the desired term, or finding the general term and then evaluating. 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.

Both approaches are shown in the following example:

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 \( 25 + 11 + 13 + 15 = 64 \).

The other approach is 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 \( 8^{\text{th}}\) term is \( 8 \times 8 = 64.\ _\square \)

Find the \( 10^{\text{th}}\) term in the sequence \( 1, 3, 5, 7, 9, \dots .\)

Looking for a pattern that allows us to find the next term, we note that each term is increased by 2. Since the initial term is 1, the general term is

\[a_n= 1 + 2\cdot (n-1).\]

Thus, the \( 10^{\text{th}}\) term is \( a_{10}=1 + 2 \cdot 9 = 19 . \ _\square \)

Find the \( 7^{\text{th}}\) term in the sequence \( 2, 4, 8, 16, 32, \dots .\)

Observe that the \(n^{\text{th}}\) term is 2 to the power of \(n,\) that is, \( a_n=2^{n} .\)

Then the \( 7^{\text{th}}\) term is \(a_7=2^{7} = 128. \ _\square \)

Find the \( 6^{\text{th}}\) term in the sequence \( 1, 1, 2, 4, 8, \dots .\)

Given the initial term 1, going forward, each term is the sum of all the previous terms. That is, the second term is 1 because 1 is the only term ahead. The third term is the sum of the first and second terms, i.e. \( 1+1 = 2.\) The \( 4^{\text{th}}\) term is the sum of all the previous terms, i.e. \( 1+1+2=4.\) Finally, the \( 5^{\text{th}}\) term is the sum of all the previous terms, which is \( 1+1+2+4=8.\)

Therefore, the \( 6^{\text{th}}\) term is the sum of all its predecessors, which is

\[ 1+1+2+4+8=16. \ _\square\]

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

\[27 \cdot \left( \frac{1}{3}\right)^{9} = \frac{3^3}{3^{9}}=\frac{1}{3^6} . \ _\square \]

Find the \( 4^{\text{th}}\) term in the sequence \( 45, 41 , 37, \dots .\)

We have

\[\begin{align} 45 - 4&=41\\ 41 - 4&=37\\ 37 - 4&=33. \end{align}\]

The \( 4^{\text{th}}\) term is 33 because in every row the difference is 4. \(_\square\)

Which of the following could describe the sequence 4, 8, 12, 16, …?

## See Also

**Cite as:**Finding Specific Terms in a Pattern.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/pattern-recognition-specific-term-2/