a_{n = ar^n-1}

Second term = 9

n = 2

ar^2-1 = 9

ar = 9 ------ eqn 1

Fourth term = 81

n = 4

ar^4-1 = 81

ar³ = 81 ----- eqn 2

Get rid of a by dividing ----- eqn 2 by ----- eqn 1

$\frac{\mathrm{a}{\mathrm{r}}^3}{\mathrm{ar}}$ = $\frac{81}{9}$

Notes:

1. The a cancel each other

2. From law of indices a^m ÷ aⁿ = $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

3. r = r¹

r^{3 - 1} = 9

r² = 9

9 = 3 x 3 = 3²

r² = 3²

Since the powers are the same, the bases are also the same.

r = 3

∴ common ratio = 3

Geometric Progression (G.P) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio.

When each succeeding terms is divided by the preceding terms, the result (common ratio) is the same.

x ÷ $\frac{16}{9}$ = $\frac{1}{\mathrm{x}}$ = $\frac{\mathrm{y}}{1}$

Using $\frac{1}{\mathrm{x}}$ = $\frac{\mathrm{y}}{1}$

Cross multiply.

x x y = 1 x 1

xy = 1

Th nth term is given by a_n = ar^{n - 1}

Where a = first term, r = common ratio

a = 3

a₅ = 3 x r^{5 - 1}

a₅ = 3r⁴

But a₅ = 48

3r⁴ = 48

Divide both sides by 3

r⁴ = $\frac{48}{3}$

r⁴ = 16

16 = 2 x 2 x 2 x 2 = 2⁴

r⁴ = 2⁴

Since the powers are the same, the bases are also the same.

r = 2

nth term = ar^n-1

First term (a) = $\frac{1}{2}$

Common ratio (r) = $\frac{{\mathrm{a}}_{\mathrm{n}}}{{\mathrm{a}}_{\mathrm{n\; -\; 1}}}$

Common ratio (r) = $\frac{1}{4}$ ÷ $\frac{1}{2}$

Reciprocate the fraction at the right of the ÷ and change ÷ to multiplication (x)

Note: reciprocate means the numerator becomes the denominator and the denominator becomes the numerator.

Common ratio (r) = $\frac{1}{4}$ x $\frac{2}{1}$ = $\frac{1}{2}$

Note: 2 divides itself one time and 4, 2 times.

nth term = ar^n-1

nth term = $\frac{1}{2}$ x ${\left(\frac{1}{2}\right)}^{\mathrm{n\; -\; 1}}$

$\frac{1}{2}$ = ${\left(\frac{1}{2}\right)}^1$ (Every number is raised to the power 1 but not written)

Law of multiplication of indices

a^m x aⁿ = a^{m + m}

nth term = ${\left(\frac{1}{2}\right)}^{\mathrm{1\; +\; n\; -\; 1}}$

nth term = ${\left(\frac{1}{2}\right)}^{\mathrm{n}}$

${\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{\mathrm{c}}$ = $\frac{{\mathrm{a}}^{\mathrm{c}}}{{\mathrm{b}}^{\mathrm{c}}}$

nth term = $\frac{1^{\mathrm{n}}}{2^{\mathrm{n}}}$

Note:
1. 1 raised to the power of any number is 1
2. 1ⁿ = 1

nth term = $\frac{1}{2^{\mathrm{n}}}$

Geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio

Common ratio = $\frac{{\mathrm{a}}_{\mathrm{n}}}{{\mathrm{a}}_{\mathrm{n\; -\; 1}}}$

Common ratio = $\frac{\mathrm{Fourth\; term}}{\mathrm{Third\; term}}$ = $\frac{\mathrm{Third\; term}}{\mathrm{Second\; term}}$ = $\frac{\mathrm{Second\; term}}{\mathrm{First\; term}}$

First term = x

Second term = 4

Third term = 16

Fourth term = y

$\frac{\mathrm{y}}{16}$ = $\frac{16}{4}$ = 4

$\frac{\mathrm{y}}{16}$ = 4

Multiply both sides by 16

y = 4 x 16 = 64

$\frac{4}{\mathrm{x}}$ = $\frac{16}{4}$ = 4

$\frac{4}{\mathrm{x}}$ = 4

Multiply both sides by x

x x 4 = 4

Divide both sides by 4

x = $\frac{4}{4}$ = 1

x : y = 1 : 64