I apologise if this comes across as being rude or blunt, but based on the long list of questions you have presented, I’d say that you have a gap in your understanding of series and sequences. I suggest you revise the main theory and formulae and redo some of the examples you did in class, then try attempting these questions.
Anyway, I’ll provide you with some guidance on how to do problems that involve finding the common difference and I’ll reference the solution of 1a in my
explanation
Let’s first look at this simple example, 1+3+5+7+9+11+13
It’s quite obvious that we are adding 2 each time, so 2 becomes our common difference. In maths we like to generalise things, so how can I mathematically find the common difference and then come up with a general formula ? Well It turns out it’s actually simple, all I need is any 3 consecutive terms of the series. For example3,5 and 7. The reason I did that is because I need to ensure that the common difference is actually ‘common’ for the whole series. So 7-5=5-3=2
So to generalise this, so it can work for any AP, we’ll re-write it like this; T(n+1)-T(n)=T(n)-T(n-1) where T(n) denotes the nth term of the series.
Now if the series is specifed to be an arithemtic series in the question, then you only need to take two consecutive terms, so in the example I’ve provided, we can deduce that from 7-5 or 9-7 or 5-3, etc..
I should also mention that there is a difference between series and sequences, a series is basically something like the example I’ve provided where as a sequence is just an ordered set of numbers, so 1,3,5,7,9,... is an example f a sequence
Now, your question states that what you’re dealing with is an AP, so what you need to do is to take any two consecutive terms,
For example, let’s take log_a (54) and log_a (18)
To find the common difference, we’ll use T(2)-T(1)=log_a(18) - log_a (54)
This can be simplified using the log rules, so it becomes log_a(18/54) which is log_a(1/3) and that’s the common difference.
Now to find a formula for the nth term, we should recall this: T(n)= a+(n-1)*d where a is the first term in the series and d is the common difference.
We just found the common difference and we know what the first term is, so all you have to is substitute them into our formula and simplify futher using the log rules.
If this type of guided explanation seems to help you, then maybe you should watch a couple of YouTube videos that explain series and sequences using a step by step approach.