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

1. the total number of Space Pods, $\mathrm{A}$, in iteration $\mathrm{n}$, and
2. the total number of Space Panels, $\mathrm{n}$, in iteration $\mathrm{n}$.

Solution:

1. 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 $\mathrm{A = n}$

1. 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}

Continue Learning
Basic Geometry Linear Equations
Number Patterns Percentage
Prime Numbers Ratio, Rate And Speed
Functions & Linear Graphs 1 Integers, Rational Numbers And Real Numbers
Basic Algebra And Algebraic Manipulation 1 Approximation And Estimation
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