What is the nth term and the next two terms of the following: (a) 0^1,2^2,24^3,252^4. (b) 1,1,2,4,7,13,24. (c) 0,1,2,3,6,11
To find the nth term of a sequence, you need to identify the pattern or rule governing the sequence. Let's analyze each sequence to find the nth term and the next two terms:
(a) To find the pattern for the sequence 0^1,2^2,24^3,252^4:
0^1 = 0^1 = 0
2^2 = 2 * 2 = 4
24^3 = 24 * 24 * 24 = 13,824
252^4 = 252 * 252 * 252 * 252 = 67,201,664
The pattern here is that each term is the exponentiated result of the previous number raised to the current term number. So, the nth term would be calculated as TN = (N-1)^(N-1), where N is the term number.
Therefore, the nth term of this sequence is (N-1)^(N-1).
To find the next two terms, substitute N with 5 and 6:
4^4 = 4 * 4 * 4 * 4 = 256
256^5 = 256 * 256 * 256 * 256 * 256 = 1,208,925,819,614,629,174,706,176
Hence, the next two terms are: 256 and 1,208,925,819,614,629,174,706,176.
(b) To find the pattern for the sequence 1,1,2,4,7,13,24:
The pattern in this sequence is not immediately obvious. Let's observe the differences between consecutive terms:
1, 2, 4, 7, 13, 24
+1, +2, +3, +6, +11
By comparing the differences, we can see that they are increasing by 1, then 2, then 3, and so on. This indicates that the pattern might be related to adding consecutive numbers.
To verify this, let's calculate the differences between these differences (also called the second differences):
1, 2, 3, 6, 11
+1, +1, +3, +5
The second differences are not constant. Therefore, the pattern is more likely to involve a quadratic equation.
By observing carefully, we can deduce that the nth term might be (n^2 - n + 1). Let's verify this by substituting values to find the next two terms:
For n = 7:
7^2 - 7 + 1 = 49 - 7 + 1 = 43
For n = 8:
8^2 - 8 + 1 = 64 - 8 + 1 = 57
Thus, the next two terms are 43 and 57.
(c) To find the pattern for the sequence 0,1,2,3,6,11:
The differences between consecutive terms are: 1, 1, 1, 3, 5.
This indicates that the pattern is likely related to adding consecutive numbers, just like in the previous sequence.
By observing carefully, we can deduce that the nth term might be (n^2 - n).
To find the next two terms, substitute n with 6 and 7:
For n = 6:
6^2 - 6 = 36 - 6 = 30
For n = 7:
7^2 - 7 = 49 - 7 = 42
Hence, the next two terms are 30 and 42.