SOLUTION: 8, a, 14, b, 20,….. the first term of the sequence above is 8. Which of the following could be the formula for finding the nth term of this sequence for any positive integer

Hmmm?  Are you talking about this sequence:

8, 14, 20, ...  ?

If so, why did you put the letter "a" in between the 8 and the 14, 
and the letter "b" between the 14 and the 20?

But, after looking at the choices given, I think you must be asking
about the sequence 8, 14, 20, ... without those letters in between
them, right?  I will assume that's what you are asking.

So let's substitute 1, 2, and 3 in each of those choices and see
which one gives you all three of those given numbers 8, 14 and 20:   

Let's try choice (a)

  2n+6   

Let's substitute n=1, n=2, and n=3 in 2n+6

2(1)+6 = 2+6 = 8 
2(2)+6 = 4+6 = 10
2(3)+6 = 6+6 = 12

Therefore, choice (a) is not the correct choice 
because it gives the sequence

8, 10, 12, ... not the sequence 8, 14, 20, ...

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Let's try choice (b)

  3n+5   

Let's substitute n=1, n=2, and n=3 in 3n+5

3(1)+5 = 3+5 = 8 
3(2)+5 = 6+5 = 11
3(3)+5 = 9+5 = 14

Therefore, choice (b) is not the correct choice 
because it gives the sequence
8, 11, 14, ... not the sequence 8, 14, 20 ... 

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Let's try choice (c)

  5n+3   

Let's substitute n=1, n=2, and n=3 in 5n+3

5(1)+3 = 5+3 = 8 
5(2)+3 = 10+3 = 13
5(3)+3 = 15+3 = 18

Therefore, choice (c) is not the correct choice 
because it gives the sequence

8, 13, 18, ... not the sequence 8, 14, 20 ... 

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Let's try choice (d)

  6n+2   

Let's substitute n=1, n=2, and n=3 in 6n+2

6(1)+2 = 6+2 = 8 
6(2)+2 = 12+2 = 14
6(3)+2 = 18+2 = 20

AHA! choice (d) IS the correct choice 
because it gives the sequence

8, 14, 20, ... 

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We have found the correct choice to be (d), so we do not really need 
to try choice (e).  But just for fun let's see why (e) is incorrect, 
like we did (a), (b) and (c):

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Trying choice (e)

  6n+5   

Let's substitute n=1, n=2, and n=3 in 6n+5

6(1)+5 = 6+5 = 11 
6(2)+5 = 12+5 = 17
6(3)+5 = 18+5 = 23

Therefore, choice (e) is not the correct choice 
because it gives the sequence

11, 17, 23, ... not the sequence 8, 14, 20 ... 

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Edwin