The following are the first four terms of a sequence: a, a (1/b),a(1/b(1/b),a(1/b)(1/b)(1/b)
where a is a positive integer. Is the eight term positive or negative? explain
So the sequence could be seen as
{a/b^(n-1): n=1,2,3...}
I hope you can see that the sign of each term will depend on the sign of b. If b is negative, then the signs for the terms will alternate.
You should also see that the odd terms of the sequence have b to an even power and the even ones have it to and odd power. Thus the answer would be whatever the sign of b is for the 8-th term. Does this make sense?
please could u tell me the nth term to term rule in 0,1,3,6,10,15,21
Thank very much
To find the nth term rule for the sequence 0, 1, 3, 6, 10, 15, 21, we need to look for a pattern in the sequence. If we observe the differences between consecutive terms, we can see that the differences are increasing by 1 each time: 1-0 = 1, 3-1 = 2, 6-3 = 3, and so on.
This pattern suggests that the differences between the terms are following a sequence of consecutive positive integers. To confirm this, we can calculate the differences of the differences:
(3-1) - (1-0) = 2 - 1 = 1,
(6-3) - (3-1) = 3 - 2 = 1, and so on.
The differences of the differences are also 1 each time, indicating that the sequence of differences is a linear sequence with a common difference of 1.
Now, to find the nth term rule, we can use this information and work backwards from the last term. Let's start with the last term, which is 21. We subtract the difference of 6 from it to get the previous term, then subtract the difference of 5 from that term, and so on, until we reach the first term:
21 - 6 = 15,
15 - 5 = 10,
10 - 4 = 6,
6 - 3 = 3,
3 - 2 = 1,
1 - 1 = 0.
At each step, we subtract the next smaller difference from the current term until we reach the first term, which is 0.
Therefore, the rule for the nth term of the sequence 0, 1, 3, 6, 10, 15, 21 is n(n-1)/2.
I hope this helps!