SG 
VN 
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, \(\mathrm{T_n}\)
For any number sequence where each term differs from the successive term by a constant quantity,
\(\mathrm{T_n = a + d(n - 1)}\)
Where, \(\mathrm{T_n}\) is the \(\mathrm{n^{th}}\) term in the sequence,
\(\mathrm{a}\) is the \(\mathrm{1^{st}}\) term in the sequence, and
\(\mathrm{d}\) 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:
\(\mathrm{1^{st}\;term}\) \(\mathrm{2^{nd}\;term}\) \(\mathrm{3^{rd}\;term}\) \(\mathrm{4^{th}\;term}\) \(\mathrm{5^{th}\;term}\)
For this sequence, it starts with \(1\), then \(+3\) to each term to get the next term.
\(\begin{align*} \mathrm{T_n} &= \mathrm{1 + 3 (n-1)}\\[2ex] &=\mathrm{1 + 3n - 3}\\[2ex] &=\mathrm{3n-2} \end{align*}\)
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 |
|---|---|---|
| \(1 \) | \(1 \) | \(4\) |
| \(2 \) | \(2 \) | \(4 + 3 = 7\) |
| \(3 \) | \(3 \) | \(4 + 3 + 3 = 10\) |
| \(4 \) | \(4 \) | \(4 + 3 + 3 + 3 = 13\) |
| \(5 \) | \(5 \) | \(4 + 3 \;(4) = 16\) |
| \(6 \) | \(6 \) | \(4 + 3 \;(5) = 19\) |
| ⋮ | ⋮ | ⋮ |
| \(\mathrm{n}\) | \(\mathrm{n}\) | \(\begin{align} \mathrm{4 + 3(n - 1)} &= \mathrm{4 + 3n - 3}\\ &= \mathrm{3n + 1} \end{align}\) |
Hence, the number of lines in the \(\mathrm{n^{th}}\) figure, \(\mathrm{L_n = 3n + 1}\)
Example 2:
The diagram shows some patterns made from floor tiles.
Find an expression, in terms of \(\mathrm{n}\), for the total number of tiles, \(\mathrm{T_n}\), 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 \(\mathrm{n}\),
\(\begin{align} \mathrm{T_n = \frac{n \;(n+1)}2} \end{align}\)
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 \(\mathrm{n}\), for
- the total number of Space Pods, \(\mathrm{A}\), in iteration \(\mathrm{n}\), and
- the total number of Space Panels, \(\mathrm{n}\), in iteration \(\mathrm{n}\).
Solution:
- In Iteration 1, there is \(1\) Space Pod.
In Iteration 2, there are \(2\) Space Pods and so on.
Hence, in Iteration \(\mathrm{n}\), the number of Space Pods would be \(\mathrm{A = n}\)
- 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 \(\mathrm{n}\), the number of Space Panels would be
\(\begin{align} \mathrm{B} &= \mathrm{4 + 2(n -1)}\\ &= \mathrm{4 + 2n - 2}\\ &= \mathrm{2n + 2} \end{align}\)
Primary
Secondary
