Understanding Nth Term in Number Sequences

[tomtomtom1]

Hi all

I'm studying sequences and series, the problem I am having is this:-

Given the sequence 8 11 14

Find the Nth Term.

I have worked out the nth term to be 3N+5, so if I wanted to find the 4th term it would be 3*4+5 = 17.

The problem is that I have come across the following formula:-

a+(n-1)d where a = 1st term, N is the Nth term, d common difference.

If I plug in the values into the above formula I get:-
8+(4-1)*3 = 17

I get the same result?? the problem I am having is that I do not understand how the two are related?

Can someone explain??

 

[haruspex]

a+(n-1)d where a = 1st term, N is the Nth term, d common difference.

Try expanding the term in parentheses and writing the result in the form A*n+B.

 

[Mentallic]

Our sequence is
a, a+d, a+2d, ...

which is equivalent to

a+0d, a+1d, a+2d, ...

So what we have is:
1st = a+0d
2nd = a+1d
3rd = a+2d
...
nth = a+(n-1)d

So in order to use a formula that's of this form, we need to make our 1st term (n=1) have a value of 0, n=2 have a value of 1... our nth value have a value of (n-1).

Hence a+(n-1)d does just this.