Find the nth term of a sequence whose first several terms are given.

6, 8, 6, 8, 6, 8, . . .

Well, this sequence seems to be caught in a clowns' cycle! It's the classic "6, 8, 6, 8, 6, 8" dance. To find the nth term, let's analyze it a bit. It alternates between 6 and 8. So, if n is odd, the term will be 6, and if n is even, it will be 8. So, we can express it as follows:

nth term = 6 if n is odd
nth term = 8 if n is even

That's how the clown keeps switching between 6 and 8 in this sequence. Hope that brings a smile to your face!

The given sequence alternates between 6 and 8. Therefore, the nth term is 6 when n is odd, and 8 when n is even.

To find the nth term of the sequence 6, 8, 6, 8, 6, 8, ..., we can observe that the pattern alternates between 6 and 8.

Since the pattern repeats every two terms, we can use modular arithmetic to express this pattern mathematically:

If n mod 2 is 0, then the term is 8.
If n mod 2 is 1, then the term is 6.

In other words, if n is even, the term is 8, and if n is odd, the term is 6.

So, if you want to find the nth term of this sequence, you can use the following formula:

If n mod 2 is 0, then the nth term is 8.
If n mod 2 is 1, then the nth term is 6.

For example:
- The first term (n=1) is odd, so the first term is 6.
- The second term (n=2) is even, so the second term is 8.
- The third term (n=3) is odd, so the third term is 6.
- And so on.

Therefore, the nth term of this sequence can be determined by checking whether n is even or odd, and assigning the appropriate value accordingly.

an = ( - 1 ) ^ n + 7

For odd numbers 1 , 3 , 5 ...

( - 1 ) ^ n = - 1

and

( - 1 ) ^ n + 7 = - 1 + 7 = 6

For even numbers 2 , 4 , 6 ...

( - 1 ) ^ n = 1

and

( - 1 ) ^ n + 7 = 1 + 7 = 8