Number Patterns
In this chapter, we will be discussing the below mentioned topics in detail:
- Common Difference (Direct)
- Common Difference (Indirect)
- General Term
- Find a formula for general term given a number pattern
- Find n given a particular iteration in the pattern
- Determine whether a given iteration is part of a number pattern
Formula for the General Term, \(T_n\)
For any number sequence where each term differs from the successive term by a constant quantity,
\(T_n = a + d \;(n\; – 1)\)
Where, \(T_n\) is the \(n^{th}\) term in the sequence,
\(a\) is the \(1^{st}\) term in the sequence, and
\(d\) is the common difference in quantity between successive terms.
Let’s understand this with the help of some examples:
Example 1:
Consider the following number sequence:
\(1^{st}\;term\) \(2^{nd}\;term\) \(3^{rd}\;term\) \(4^{th}\;term\) \(5^{th}\;term\)
For this sequence, it starts with \(1\), then \(+3\) to each term to get the next term.
\(\begin{align*} T_n &= 1 + 3 (n\; – 1)\\ &=1 + 3 n\; – 3\\ &=3n-2 \end{align*}\)
Number Patterns
A number pattern is a sequence of figures linked by a specific rule.
Example 1:
Consider the following number pattern:
How would the next two figures look like?
Solution:
Figure Number |
Number Of Squares |
Number Of Lines |
|---|---|---|
| \(1 \) | \(1 \) | \(4\) |
|
\(2 \) |
\(2 \) |
\(4 + 3 = 7\) |
|
\(3 \) |
\(3 \) |
\(4 + 3 + 3 = 10\) |
|
\(4 \) |
\(4 \) |
\(4 + 3 + 3 + 3 = 13\) |
|
\(5 \) |
\(5 \) |
\(4 + 3 \;(4) = 16\) |
|
\(6 \) |
\(6 \) |
\(4 + 3 \;(5) = 19\) |
|
⋮ |
⋮ |
⋮ |
|
\(n \) |
\(n \) |
\(\begin{align*} 4 + 3(n – 1) &= 4 + 3n – 3\\ &= 3n + 1 \end{align*}\) |
Hence, the number of lines in the \(n^{th}\) figure, \(L_n = 3n + 1\)
Example 2:
The diagram shows some patterns made from floor tiles.
Find an expression, in terms of \(n\), for the total number of tiles, \(T_n\), in Figure \(n\).
Solution:
Figure Number |
Number Of Tiles |
|---|---|
|
\(1\) |
\(\begin{align*} 1 && && && && =1 && && = \frac{1\times2}{2} \end{align*}\) |
|
\(2 \) |
\(\begin{align*} 1+2 && && && =3 && = \frac{2\times3}{2} \end{align*}\) |
|
\(3 \) |
\(\begin{align*} 1+2+3 && && =6 && = \frac{3\times4}{2} \end{align*}\) |
|
\(4 \) |
\(\begin{align*} 1+2+3+4 &&=10 && = \frac{4\times5}{2} \end{align*} \) |
In Figure \(n\),
\(T_n = \frac{n \;(n+1)}2\)
Question 1:
The International Space Station (ISS) consists of oval shaped Space Pods and rectangular Solar Panels. The first 3 iterations of the ISS are as shown.
Find a formula, in terms of \(n\), for
- the total number of Space Pods, \(A\), in iteration \(n\), and
- the total number of Space Panels, \(n\), in iteration \(n\).
Solution:
- In Iteration 1, there is \(1\) Space Pod.
In Iteration 2, there are \(2\) Space Pods and so on.
Hence, in Iteration n, the number of Space Pods would be \(A = n\)
- In Iteration 1, there are \(4\) Space Panels; in Iteration 2, there are \(6\) Space Panels; and in Iteration 3, there are \(8\) Space Panels and so on.
Hence, in Iteration n, the number of Space Panels would be
\(\begin{align*} B &= 4 + 2 \;(n \;– 1)\\ &= 4 + 2n \;– 2\\ &= 2n + 2 \end{align*}\)
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