Identify if the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic or geometric, then find the next term. and write the nth term.
. t-3,t-2, t-1, t...
. 1,-3/2,2,-5/2....
Find the fifth term and the nth term of the geometric sequence whose initial term, a, and common ratio, r, are given.
a =^3 ; r =^3
To determine if a given sequence is arithmetic, geometric, or neither, we need to examine the differences or ratios between consecutive terms.
1) For the sequence t-3, t-2, t-1, t...
We can observe that the common difference is 1, which means that each term is obtained by adding 1 to the previous term. Therefore, this sequence is arithmetic.
To find the next term, we add the common difference (1) to the last term (t). Thus, the next term is (t + 1).
To write the nth term formula, we can use the general formula for arithmetic sequences:
tn = a + (n-1)d
In this case, the first term (a) is (t - 3), and the common difference (d) is 1. Therefore, the nth term formula for this sequence is:
tn = t - 3 + (n-1)(1) or tn = t + n - 4.
2) For the sequence 1, -3/2, 2, -5/2...
We can see that the successive terms in this sequence do not have a constant difference or ratio. Therefore, this sequence is neither arithmetic nor geometric.
Since the sequence is neither arithmetic nor geometric, we cannot accurately determine the next term or write a formula for the nth term.