TL;DR

Linear, quadratic and geometric sequences can be ‘hidden’ or ‘disguised’ within an outer mathematical structure (e.g. roots, powers, fractions).
The $n$th term is the $n$th term of the inner sequence within the outer structure.

Often in the investigation, the sequence will have the structure of a linear, quadratic or geometric sequence, but be hidden or disguised within some other larger structure.

For example, consider the sequence $3,7,11,15,19,…$ which has $n$th term $4n-1$.

Now consider each of these related sequences:

The sequence $\sqrt{3},\sqrt{7},\sqrt{11},\sqrt{15},\sqrt{19},…$ has $n$th term $\sqrt{4n-1}$.
The sequence $2^{3},2^{7},2^{11},2^{15},2^{19},…$ has $n$th term $2^{4n-1}$.
The sequence $\frac{1}{3},\frac{1}{7},\frac{1}{11},\frac{1}{15},\frac{1}{19},…$ has $n$th term $\frac{1}{4n-1}$.
The sequence $2\times 10^{3},2\times 10^{7},2\times 10^{11},2\times 10^{15},2\times 10^{19},…$ has $n$th term $2\times 10^{4n-1}$.
The sequence $3^{x},7^{x},11^{x},15^{x},19^{x},…$ has $n$th term $\left(4n-1\right)^{x}$.

In all these examples, the same inner linear sequence is hidden within another outer mathematical structure, which forms part of the $n$th term.

Example

Find the $n$th term of the sequence$$-11y, -8y, -5y, -2y, y, …$$

The outer structure is $?y$.
The inner sequence is $-11, -8, -5, -2, 1, …$ which is linear with common difference $3$ and has $n$th term $3n-14$.
So the $n$th term is $$\left(3n-14\right)y$$

Example

Find the $n$th term of the sequence$$\sqrt{2}, \sqrt{18},\ sqrt{162}, \sqrt{1458}, \sqrt{13122}, …$$

The outer structure is $\sqrt{?}$.
The inner sequence is $2, 18, 162, 1458, 13122, …$ which is geometric with common ratio $9$ and has $n$th term $2\times 9^{n-1}$.
So the $n$th term is $$\sqrt{2\times 9^{n-1}}$$

Example

Find the $n$th term of the sequence$$5^{2}, 5^{8}, 5^{16}, 5^{26}, 5^{38}, …$$

The outer structure is $5^{?}$.
The inner sequence is $2, 8, 16, 26, 38, …$ which is quadratic with second difference $2$ and has $n$th term $n^{2}+3n-2$.
So the $n$th term is $$5^{n^{2}+3n-2}$$

Example

Find the $n$th term of the sequence$$\frac{5}{4}, \frac{5}{13}, \frac{5}{22}, \frac{5}{31}, \frac{5}{40}, …$$

The outer structure is $\frac{5}{?}$.
The inner sequence is $4, 13, 22, 31, 40, …$ which is linear with common difference $9$ and has $n$th term $9n-5$.
So the $n$th term is $$\frac{5}{9n-5}$$

Example

Find the $n$th term of the sequence$$\left(23.9, 22.7\right), \left(27.8, 32.4\right), \left(31.7, 42.1\right), \left(35.6,51.8\right), \left(39.5, 61.5\right), …$$

The outer structure is $\left(?, ?\right)$.
The first inner sequence is $23.9, 27.8, 31.7, 35.6, 39.5, …$ which is linear with common difference $3.9$ and has $n$th term $3.9n+20$.
The second inner sequence is $22.7, 32.4, 42.1, 51.8, 61.5, …$ which is linear with common difference $9.7$ and has $n$th term $9.7n+13$.
So the $n$th term is $$\left(3.9n+20, 9.7n+13\right)$$

Example

Find the $n$th term of the sequence$$\frac{-2}{2}, \frac{4}{9}, \frac{10}{18}, \frac{16}{29}, \frac{22}{42}, …$$

The outer structure is $\frac{?}{?}$.
The first inner sequence is $-2, 4, 10, 16, 22, …$ which is linear with common difference $6$ and has $n$th term $6n-8$.
The second inner sequence is $2, 9, 18, 29, 42, …$ which is quadratic with second difference $1$ and has $n$th term $n^{2}+4n-3$.
So the $n$th term is $$\frac{6n-8}{n^{2}+4n-3}$$

  
      Disguised Sequence
      Outer Structure
      Inner Type
      Key Details
      \(n\)the Term
    
      \( \sqrt{3}, \sqrt{8}, \sqrt{13}, \sqrt{18}, \dots \)
      \( \sqrt{x} \)?

      \( \text{Linear} \)?

      \( d=+5 \)
\( a_0=-2 \)?

      \( \sqrt{5n-2} \)?

      \( \sin(15^\circ), \sin(25^\circ), \sin(35^\circ), \dots \)
      \( \sin(x) \)?

      \( \text{Linear} \)?

      \( d=+10 \)
\( a_0=5 \)?

      \( \sin(10n+5) \)?

      \( 5^5, 5^8, 5^{11}, 5^{14}, \dots \)
      \( 5^x \)?

      \( \text{Linear} \)?

      \( d=+3 \)
\( a_0=2 \)?

      \( 5^{3n+2} \)?

      \( \sqrt[3]{2}, \sqrt[3]{4}, \sqrt[3]{8}, \sqrt[3]{16}, \dots \)
      \( \sqrt[3]{x} \)?

      \( \text{Geometric} \)?

      \( a=2 \)
\( r=2 \)?

      \( \sqrt[3]{2^n} \)?

      \( \frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \dots \)
      \( \frac{1}{x} \)?

      \( \text{Quadratic} \)?

      \( 2a=2 \)
\( u_n-an^2 = n \)?

      \( \frac{1}{n^2+n} \)?

      \( (1, 2), (2, 5), (3, 10), (4, 17), \dots \)
      \( (n, f(n)) \)?

      \( \text{Quadratic } y \)?

      \( y = n^2+1 \)?

      \( (n, n^2+1) \)?

      \( \begin{pmatrix} 2 \\ 5 \end{pmatrix}, \begin{pmatrix} 4 \\ 15 \end{pmatrix}, \begin{pmatrix} 8 \\ 45 \end{pmatrix}, \dots \)
      \( \text{Vector} \)?

      \( \text{Double Geo} \)?

      \( r_{top}=2 \)
\( r_{bot}=3 \)?

      \( \begin{pmatrix} 2^n \\ 5 \cdot 3^{n-1} \end{pmatrix} \)?

      \( -\pi, 2\pi, 7\pi, 14\pi, \dots \)
      \( x \cdot \pi \)?

      \( \text{Quadratic} \)?

      \( 2a=2 \)
\( u_n-an^2 = -2 \)?

      \( (n^2-2)\pi \)?

      \( 3 \times 10^1, 3 \times 10^2, 3 \times 10^3, \dots \)
      \( 3 \times 10^x \)?

      \( \text{Geometric} \)?

      \( a=30 \)
\( r=10 \)?

      \( 3 \times 10^n \)?

      \( \frac{2}{3}, \frac{4}{5}, \frac{6}{7}, \frac{8}{9}, \dots \)
      \( \frac{f(n)}{g(n)} \)?

      \( \text{Linear/Linear} \)?

      \( N: 2n \)
\( D: 2n+1 \)?

      \( \frac{2n}{2n+1} \)?

      \( 2x+1, 2x+3, 2x+5, \dots \)
      \( 2x + (\dots) \)?

      \( \text{Linear} \)?

      \( d=+2 \)
\( a_0=-1 \)?

      \( 2x + (2n-1) \)?

      \( (x+1)^2, (x+2)^2, (x+3)^2, \dots \)
      \( (\dots)^2 \)?

      \( \text{Linear Base} \)?

      \( \text{Base: } x+n \)?

      \( (x+n)^2 \)?

      \( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \dots \)
      \( \frac{1}{x} \)?

      \( \text{Linear} \)?

      \( d=+3 \)
\( a_0=-2 \)?

      \( \frac{1}{3n-2} \)?

      \( \frac{3}{6}, \frac{3}{13}, \frac{3}{24}, \dots \)
      \( \frac{3}{x} \)?

      \( \text{Quadratic} \)?

      \( 2a=4 \)
\( bn+c=n+3 \)?

      \( \frac{3}{2n^2+n+3} \)?

      \( 5^2, 5^3, 5^4, 5^5, \dots \)
      \( 5^x \)?

      \( \text{Linear} \)?

      \( d=+1 \)
\( a_0=1 \)?

      \( 5^{n+1} \)?

      \( \sqrt[5]{2}, \sqrt[10]{2}, \sqrt[15]{2}, \sqrt[20]{2}, \dots \)
      \( \sqrt[x]{2} \)?

      \( \text{Linear} \)?

      \( d=+5 \)
\( a_0=0 \)?

      \( \sqrt[5n]{2} \)?

      \( 10\pi, 5\pi, 2.5\pi, 1.25\pi, \dots \)
      \( x \cdot \pi \)?

      \( \text{Geometric} \)?

      \( a=10 \)
\( r=0.5 \)?

      \( 10(0.5)^{n-1}\pi \)?

      \( 4 \times 10^{0}, 4 \times 10^{2}, 4 \times 10^{6}, 4 \times 10^{12} \dots \)
      \( 4 \times 10^x \)?

      \( \text{Quadratic} \)?

      \( 2a=2 \)
\( u_n-an^2 = -n \)?

      \( 4 \times 10^{n^2-n} \)?

Disguised Sequence	Outer Structure	Inner Type	Key Details	\(n\)the Term
\( \sqrt{3}, \sqrt{8}, \sqrt{13}, \sqrt{18}, \dots \)	\( \sqrt{x} \) ?	\( \text{Linear} \) ?	\( d=+5 \) \( a_0=-2 \) ?	\( \sqrt{5n-2} \) ?
\( \sin(15^\circ), \sin(25^\circ), \sin(35^\circ), \dots \)	\( \sin(x) \) ?	\( \text{Linear} \) ?	\( d=+10 \) \( a_0=5 \) ?	\( \sin(10n+5) \) ?
\( 5^5, 5^8, 5^{11}, 5^{14}, \dots \)	\( 5^x \) ?	\( \text{Linear} \) ?	\( d=+3 \) \( a_0=2 \) ?	\( 5^{3n+2} \) ?
\( \sqrt[3]{2}, \sqrt[3]{4}, \sqrt[3]{8}, \sqrt[3]{16}, \dots \)	\( \sqrt[3]{x} \) ?	\( \text{Geometric} \) ?	\( a=2 \) \( r=2 \) ?	\( \sqrt[3]{2^n} \) ?
\( \frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, \dots \)	\( \frac{1}{x} \) ?	\( \text{Quadratic} \) ?	\( 2a=2 \) \( u_n-an^2 = n \) ?	\( \frac{1}{n^2+n} \) ?
\( (1, 2), (2, 5), (3, 10), (4, 17), \dots \)	\( (n, f(n)) \) ?	\( \text{Quadratic } y \) ?	\( y = n^2+1 \) ?	\( (n, n^2+1) \) ?
\( \begin{pmatrix} 2 \\ 5 \end{pmatrix}, \begin{pmatrix} 4 \\ 15 \end{pmatrix}, \begin{pmatrix} 8 \\ 45 \end{pmatrix}, \dots \)	\( \text{Vector} \) ?	\( \text{Double Geo} \) ?	\( r_{top}=2 \) \( r_{bot}=3 \) ?	\( \begin{pmatrix} 2^n \\ 5 \cdot 3^{n-1} \end{pmatrix} \) ?
\( -\pi, 2\pi, 7\pi, 14\pi, \dots \)	\( x \cdot \pi \) ?	\( \text{Quadratic} \) ?	\( 2a=2 \) \( u_n-an^2 = -2 \) ?	\( (n^2-2)\pi \) ?
\( 3 \times 10^1, 3 \times 10^2, 3 \times 10^3, \dots \)	\( 3 \times 10^x \) ?	\( \text{Geometric} \) ?	\( a=30 \) \( r=10 \) ?	\( 3 \times 10^n \) ?
\( \frac{2}{3}, \frac{4}{5}, \frac{6}{7}, \frac{8}{9}, \dots \)	\( \frac{f(n)}{g(n)} \) ?	\( \text{Linear/Linear} \) ?	\( N: 2n \) \( D: 2n+1 \) ?	\( \frac{2n}{2n+1} \) ?
\( 2x+1, 2x+3, 2x+5, \dots \)	\( 2x + (\dots) \) ?	\( \text{Linear} \) ?	\( d=+2 \) \( a_0=-1 \) ?	\( 2x + (2n-1) \) ?
\( (x+1)^2, (x+2)^2, (x+3)^2, \dots \)	\( (\dots)^2 \) ?	\( \text{Linear Base} \) ?	\( \text{Base: } x+n \) ?	\( (x+n)^2 \) ?
\( 1, \frac{1}{4}, \frac{1}{7}, \frac{1}{10}, \dots \)	\( \frac{1}{x} \) ?	\( \text{Linear} \) ?	\( d=+3 \) \( a_0=-2 \) ?	\( \frac{1}{3n-2} \) ?
\( \frac{3}{6}, \frac{3}{13}, \frac{3}{24}, \dots \)	\( \frac{3}{x} \) ?	\( \text{Quadratic} \) ?	\( 2a=4 \) \( bn+c=n+3 \) ?	\( \frac{3}{2n^2+n+3} \) ?
\( 5^2, 5^3, 5^4, 5^5, \dots \)	\( 5^x \) ?	\( \text{Linear} \) ?	\( d=+1 \) \( a_0=1 \) ?	\( 5^{n+1} \) ?
\( \sqrt[5]{2}, \sqrt[10]{2}, \sqrt[15]{2}, \sqrt[20]{2}, \dots \)	\( \sqrt[x]{2} \) ?	\( \text{Linear} \) ?	\( d=+5 \) \( a_0=0 \) ?	\( \sqrt[5n]{2} \) ?
\( 10\pi, 5\pi, 2.5\pi, 1.25\pi, \dots \)	\( x \cdot \pi \) ?	\( \text{Geometric} \) ?	\( a=10 \) \( r=0.5 \) ?	\( 10(0.5)^{n-1}\pi \) ?
\( 4 \times 10^{0}, 4 \times 10^{2}, 4 \times 10^{6}, 4 \times 10^{12} \dots \)	\( 4 \times 10^x \) ?	\( \text{Quadratic} \) ?	\( 2a=2 \) \( u_n-an^2 = -n \) ?	\( 4 \times 10^{n^2-n} \) ?

Lesson 7 - Disguised Sequences

TL;DR