Given that log𝑒 3 , log𝑒 32 , log𝑒 33 , …. form an
arithmetic progression
18. What is the common difference
A. 2
B. 3
C. log𝑒 2
D. log𝑒 3
We can use the formula for the nth term of an arithmetic progression to find the common difference:
𝑎𝑛 = 𝑎1 + (𝑛 - 1)𝑑
where 𝑎1 is the first term, 𝑑 is the common difference, and 𝑎𝑛 is the nth term.
In this case, we have 𝑎1 = log𝑒 3 and 𝑑 = the common difference we want to find.
To find 𝑎2, we add the common difference to 𝑎1:
𝑎2 = 𝑎1 + 𝑑 = log𝑒 3 + 𝑑
To find 𝑎3, we add the common difference again:
𝑎3 = 𝑎2 + 𝑑 = log𝑒 3 + 2𝑑
And so on. We can see that the nth term will be:
𝑎𝑛 = log𝑒 3 + (𝑛 - 1)𝑑
Since we're given that the terms form an arithmetic progression, we know that the difference between consecutive terms is constant. Therefore, we can find the common difference by subtracting 𝑎1 from 𝑎2:
𝑎2 - 𝑎1 = (log𝑒 3 + 𝑑) - log𝑒 3 = 𝑑
So the answer is (B) 3.
To find the common difference of an arithmetic progression, we need to find the difference between any two consecutive terms.
In this case, let's take the difference between log𝑒 33 and log𝑒 32:
log𝑒 33 - log𝑒 32 = log𝑒 (33/32)
Since the common ratio between consecutive terms is 33/32, we can see that the common difference is log𝑒 (33/32).
Therefore, the correct answer is C. log𝑒 2.