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.
1. t-3,t-2, t-1, t...
2. 3/4,1/2,1/3,2/9...
3. 1,-3/2,2,-5/2....
check for common difference or ratio:
#1: d = 1
#2: r = 2/3
#3: differences: -5/2, +7/2, -9/2, ...
ratios: -3/2, +4/3, -5/4, ...
neither d nor r is constant, so the sequence is not AP or GP
so one is arithmetic
number two is geometric
number three is neither?
my bad that was my middle name
so one and two are neither
To identify if a sequence is arithmetic, geometric, or neither, we need to look for a pattern in the differences or ratios between consecutive terms.
1. For the first sequence, t-3, t-2, t-1, t...
Here, the differences between consecutive terms are 1 (t-3 to t-2) and 1 (t-2 to t-1). Since the difference is constant, this sequence is arithmetic.
The next term can be found by adding the common difference to the last term. So, the next term would be t+1.
The nth term can be determined using the formula: tn = a + (n - 1)d, where a is the first term and d is the common difference. Here, a = t-3 and d = 1. Therefore, the nth term is t-3 + (n-1)(1), or t-2 + n-1, simplified to t+n-3.
2. For the second sequence, 3/4, 1/2, 1/3, 2/9...
Here, the ratios between consecutive terms are 0.75 (3/4 ÷ 1/2), 0.66 (1/2 ÷ 1/3), and 0.67 (1/3 ÷ 2/9). Since the ratios are not constant, this sequence is neither arithmetic nor geometric. Therefore, we cannot determine the next term or the nth term without additional information.
3. For the third sequence, 1, -3/2, 2, -5/2...
Here, the differences between consecutive terms are -2.5 (1 to -3/2), 2.5 (-3/2 to 2), and -2.5 (2 to -5/2). Since the differences are not constant, this sequence is neither arithmetic nor geometric. Therefore, we cannot determine the next term or the nth term without additional information.
In conclusion:
- The first sequence is arithmetic, with the next term being t+1, and the nth term being t+n-3.
- The second and third sequences are neither arithmetic nor geometric, so we cannot determine the next term or the nth term without more information.