(a)

U_n = ar^n-1

U₃ = ar^3-1

U₃ = ar²

But third term = 4

4 = ar² ---- eqn 1

U₈ = ar^8-1

U₈ = ar⁷

But eighth term = $\frac{1}{256}$

256 = 4⁴

From indices,

$\frac{1}{{\mathrm{a}}^{\mathrm{b}}}$ = a^-b

$\frac{1}{256}$ = $\frac{1}{4^4}$ = 4^-4

4^-4 = ar⁷ -- eqn 2

eqn 1 ÷ eqn 2

$\frac{4}{4^{-4}}$ = $\frac{\mathrm{a}{\mathrm{r}}^2}{\mathrm{a}{\mathrm{r}}^7}$

The a cancels out

$\frac{4}{4^{-4}}$ = $\frac{{\mathrm{r}}^2}{{\mathrm{r}}^7}$

From indices,

a^m ÷ aⁿ = $\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

Every number is raised to the power 1

4 = 4¹

$\frac{4}{4^{-4}}$ = 4^{1 - (-4)} = 4^{1 + 4} = 4⁵

$\frac{{\mathrm{r}}^2}{{\mathrm{r}}^7}$ = r^{2 - 7} = r^-5

4⁵ = r^-5

r^-5 = $\frac{1}{{\mathrm{r}}^5}$

4⁵ = $\frac{1}{{\mathrm{r}}^5}$

Every number is being divided by 1

4⁵ = $\frac{4^5}{1}$

$\frac{4^5}{1}$ = $\frac{1}{{\mathrm{r}}^5}$

Cross multiply.

4⁵ x r⁵ = 1 x 1

Divide both sides by 4⁵

r⁵ = $\frac{1}{4^5}$

Take the 5th root of both sides

r = ${\left(\frac{1}{4^5}\right)}^{\frac{1}{5}}$

Note:

1. 1 raised to any power is 1 hence 1 to the power $\frac{1}{5}$ is still 1

2. $4^5$ raised to the power $\frac{1}{5}$ is 4 since the 5 cancels each other

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

(b)

Subtitute the common ratio (r) into equation 1 and solve for a

4 = ar² ---- eqn 1

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

4 = a x ${\left(\frac{1}{4}\right)}^2$

4 = a x $\frac{1}{4^2}$

4 = a x $\frac{1}{16}$

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

Multiply both sides by 16

a = 4 x 16 = 64

First term = 64

(c)

U₂ = ar^2-1

U₂ = ar¹

U₂ = ar

a = 64

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

Second term = 64 x $\frac{1}{4}$ = 16

first term + second term = 64 + 16

first term + second term = 80

Alternately

S_n = a$\left[\frac{{\mathrm{r}}^{\mathrm{n}}\mathrm{-\; 1}}{\mathrm{r\; -\; 1}}\right]$ when r > 1

S_n = a$\left[\frac{\mathrm{1\; -}{\mathrm{r}}^{\mathrm{n}}}{\mathrm{1\; -\; r}}\right]$ when r < 1

r = $\frac{1}{4}$ and r < 1

a = 64

S₂ = 64 x $\left[\frac{\mathrm{1\; -}{\left(\frac{1}{4}\right)}^2}{\mathrm{1\; -}\frac{1}{4}}\right]$

S₂ = 64 x $\left[\frac{\mathrm{1\; -}\frac{1}{4^2}}{\frac{1}{1}\mathrm{-}\frac{1}{4}}\right]$

S₂ = 64 x $\left[\frac{\frac{1}{1}\mathrm{-}\frac{1}{16}}{\frac{\mathrm{4\; -\; 1}}{4}}\right]$

S₂ = 64 x $\left[\frac{\frac{\mathrm{16\; -\; 1}}{16}}{\frac{3}{4}}\right]$

S₂ = 64 x $\left[\frac{\frac{15}{16}}{\frac{3}{4}}\right]$

S₂ = 64 x $\frac{15}{16}$ ÷ $\frac{3}{4}$

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

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

S₂ = 64 x $\frac{15}{16}$ x $\frac{4}{3}$

Note:
1. 4 divides 16, 4 times
2. 3 divides 15, 5 times
3. 4 divides 64, 16 times

S₂ = 16 x 5 = 80