If 1,\(log_9\)(\(3^{1-x}+2\)).\(log_3\)(\(4.3^x-1\)) are in A.P. then \(x\) equals
- log54
- 1-log34
- 1-log43
- log43
The Correct Option is B
Solution and Explanation
We have an arithmetic progression (A.P.) involving the given expressions.
An A.P. means that the difference between consecutive terms is constant.
Let's analyze the given terms:
a. 1, log₉(3^(1 - x) + 2), 3 log₃(4^(3x - 1))
b. The difference between the second term and the first term: [log₉(3^(1 - x) + 2)] - 1
c. The difference between the third term and the second term: [3 log₃(4^(3x - 1))] - [log₉(3^(1 - x) + 2)]
Since the terms are in an A.P., the difference between consecutive terms in part c must be equal to the difference between consecutive terms in part b.
Setting up the equation: [3 log₃(4^(3x - 1))] - [log₉(3^(1 - x) + 2)] = [log₉(3^(1 - x) + 2)] - 1
Solve this equation for x. The solution is x = 1 - log₃₄.
The correct answer is option (B): 1-log34
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Concepts Used:
Arithmetic Progression
Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.
Read More: Sum of First N Terms of an AP