what is the nth term of the following sequence:
1, -1/2, 3/4, -15/8, etc
To find the nth term of a sequence, we need to identify any patterns or rules that govern the sequence. In this case, we notice that each term in the sequence is obtained by multiplying the previous term by a constant factor. Let's break it down step by step:
1st term: 1
2nd term: -1/2 (multiplied the 1st term by -1/2)
3rd term: 3/4 (multiplied the 2nd term by 3/2)
4th term: -15/8 (multiplied the 3rd term by -5/2)
We can see that to obtain each term, we are multiplying by a factor that alternates between -1/2 and -5/2. Specifically, the factor is -1/2 for odd terms and -5/2 for even terms.
To express this pattern mathematically, we can use the following formula for the nth term:
nth term = ((-1)^(n+1) * 2^(n-1)) / 2^n
Using this formula, we can find any term in the sequence by substituting n with the desired term number. For example, if we want to find the 5th term:
5th term = ((-1)^(5+1) * 2^(5-1)) / 2^5
= ((-1)^6 * 2^4) / 2^5
= (1 * 16) / 32
= 16/32
= 1/2
So, the 5th term of the sequence is 1/2.