find the rule for the Nth term of the arithmetic sequence. 11/2, 25/6, 17/6, 3/2, 1/6.....
If you change the denomators to 6, you should notice the numerators follow the sequence:
33,25,17,9,1,...which is an arithmetic sequence with a common difference of -8
so term(n) = -8n + k
since for n=1 term(1)=33, k must be 41
so term(n) = -8n + 41
To find the rule for the nth term of the arithmetic sequence, we need to determine the pattern followed by the sequence.
Looking at the numerators, we can observe that they form an arithmetic sequence of 33, 25, 17, 9, 1. The common difference between each term in the arithmetic sequence is -8.
Now, we can write the rule for the nth term as follows: term(n) = -8n + k.
To determine the value of k, we need to find the value of term(1).
When n = 1, the first term is 11/2. However, we notice that the given sequence has fractions with different denominators. So, we need to change the denominators to a common value, such as 6.
Multiplying each term by 6, we get the new sequence: 33/6, 25/6, 17/6, 9/6, 1/6.
Now, we can see that the numerator sequence follows the arithmetic sequence we identified earlier. The first term of the numerator sequence is 33/6.
Taking the common denominator, we have term(1) = 33/6.
So, when n = 1, term(1) = 33/6.
We can substitute these values into the rule equation to solve for k:
33/6 = -8(1) + k
Now, simplifying the equation:
33/6 = -8 + k
Multiplying both sides by 6:
33 = -48 + 6k
Adding 48 to both sides:
33 + 48 = 6k
81 = 6k
Dividing both sides by 6:
81/6 = k
13.5 = k
Therefore, k is equal to 13.5.
Finally, we can substitute the value of k into the rule equation:
term(n) = -8n + 13.5.
So, the rule for the nth term of the arithmetic sequence is -8n + 13.5.