Question

[4 marks

Answer 1

Answer:

1. x = 3

2. The 5th term = Log₂768.

Step-by-step explanation:

From the question given above, the following data were obtained:

Log₂x, Log₂(x + 9, Log₂(x + 45)

x =?

5th term (T₅) =.?

1. Determination of the value of x.

We shall determine the value of x as follow:

First term = Log₂x

2nd term = Log₂(x + 9)

3rd term = Log₂(x + 45)

Common difference = 2nd term – first term = 3rd term – 2nd term

Log₂(x + 9) – Log₂x = Log₂(x + 45) – Log₂(x + 9)

Recall

Log M – Log N = Log (M/N)

Therefore,

Log₂(x + 9) – Log₂x = Log₂(x + 9)/x

Log₂(x + 45) – Log₂(x + 9) = Log₂(x + 45) /(x + 9)

Thus:

Log₂(x + 9) – Log₂x = Log₂(x + 45) – Log₂(x + 9)

Log₂(x + 9)/x = Log₂(x + 45)/(x + 9)

Cancel Log₂ from both side

(x + 9)/x = (x + 45)/(x + 9)

Cross multiply

(x + 9)(x + 9) = x(x + 45)

x² + 9x + 9x + 81 = x² + 45x

x² + 18x + 81 = x² + 45x

Rearrange

x² – x² + 81 = 45x – 18x

81 = 27x

Divide both side by 27

x = 81/27

x = 3

Therefore, the value of x is 3.

2. Determination of the 5th term.

We'll begin by calculating the common difference (d)

x = 3

First term = Log₂x

First term = Log₂3

2nd term = Log₂(x + 9)

2nd term = Log₂(3 + 9)

2nd term = Log₂12

Common difference (d) = 2nd term – first term

Common difference (d) = Log₂12 – Log₂3

= Log₂(12/3)

Common difference (d) = Log₂4

Finally, we shall determine the 5th term as follow:

First term (a) = Log₂3

Common difference (d) = Log₂4

5th term (T₅) =.?

T₅ = a + 4d

T₅ = Log₂3 + 4Log₂4

Recall:

nLogM = LogMⁿ

Therefore,

4Log₂4 = Log₂4⁴ = Log₂256

T₅ = Log₂3 + 4Log₂4

T₅ = Log₂3 + Log₂256

Recall:

Log M + Log N = Log (M×N)

Thus,

T₅ = Log₂3 + Log₂256

T₅ = Log₂(3 × 256)

T₅ = Log₂768

Therefore, the 5th term is Log₂768