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
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