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1.Let the first term be a and the common ratio r
We know the first term a = 5
Use the formula for the n'th term: x_n = ar^{n − 1}
The sixth term = 160 ⇒ x₆ = ar^{6 − 1} = 160 ⇒ ar⁵ = 160
But a = 5
Therefore 5r⁵= 160 ⇒ r⁵= 160 ÷ 5 = 32

2. Let the first term be a and the common ratio r

Use the formula for the n'th term: x_n = ar^{n − 1}

The fourth term = 27 ⇒ x₄ = ar^{4 − 1} = 27 ⇒ ar³ = 27 (1)

The seventh term = 1 ⇒ x₇ = ar^{7 − 1} = 1 ⇒ ar⁶ = 1 (2)

Divide (2) by (1) ⇒ r³ = 1/27 ⇒ r = 1/3

Substitute r = 1/3 into (1) ⇒ a × (1/3)³ = 27 ⇒ a × 1/27 = 27
⇒ a = 27 × 27 = 729

The first term is 729

3. Let the first term be a and the common difference d
Use the formula for the nth term: x_n = a + d(n − 1)

The first term = 4 ⇒ a = 4 (1)
The tenth term = 67 ⇒ x₁₀ = a + d(10 − 1) = 67 ⇒ a + 9d = 67 (2)

Substitute a = 4 from (1) into (2)
⇒ 4 + 9d = 67
⇒ 9d = 63
⇒ d = 63 ÷ 9 = 7

The common difference is 7

4. This sequence has a difference of 5 between each pair of numbers.

The values of a and d are:
a = -12 (the first term)
d = 5 (the "common difference")

The Rule can be calculated:
x_n = a + d(n − 1)
= -12 + 5(n − 1)
= -12 + 5n − 5
= 5n − 17

So, the 32nd term is:
x₃₂ = 5 × 32 − 17 = 160 − 17 = 143