Find the general term in simplest form for the sequence 2,1,-4,7,-10,13,-16.
There's an addition of 3 with alternating signs I think but I'm unable to find a general term.
suppose they were all positive:
2,1,4,7,10,13,16
that general term would be 3n-5, from the second term on
It looks like the first term is an outlier if I make that change, let's hope it works out
but we want to make the terms alternate between + and -
this can be done by (-1)^n
so the sequence can be defined as follows
term(n) = (-1)^n (3n-5)
testing:
term(1) = (-1)^1 (3 - 5) = -1(-2) = 2
term(2) = (-1)^2 (6-1) = (=1)(1) = 1
....
term(5) = (-1)^5 (15-5) = (-1)(10) = -10
term(6) = (-1)^6 (18-5) = (+1)(13 = 13
looks ok
Thank you
To find the general term for this sequence, we can observe that the sequence alternates between adding 3 and subtracting 6. Let's break it down:
2 + 3 = 5
5 - 6 = -1
-1 + 3 = 2
2 - 6 = -4
-4 + 3 = -1
-1 - 6 = -7
-7 + 3 = -4
-4 - 6 = -10
-10 + 3 = -7
-7 - 6 = -13
-13 + 3 = -10
-10 - 6 = -16
By examining the pattern, we can see that the terms with odd indices are generated by adding 3 each time, while the terms with even indices are generated by subtracting 6.
We can express the general term for the sequence as follows:
aₙ = ((-1)^(n+1) * 3 * (n-1))/2
where n represents the position of the term in the sequence.
So, the general term in simplest form for the given sequence is ((-1)^(n+1) * 3 * (n-1))/2.
To find the general term for this sequence, let's examine the pattern more closely. We can see that the sign is alternating, with +, -, +, -, +, - in each consecutive term.
Now, let's look at the numbers themselves. It appears that the numbers themselves are increasing by 3 each time, but with alternating signs.
To express this pattern algebraically, we can start with the first term and add (or subtract) a fixed number each time. In this case, let's add 3 to the first term, then subtract 3, and so on.
Let n be the position of the term in the sequence (starting with n = 1).
If n is even, we add 3:
Term(n) = 2 + 3(n-1)
If n is odd, we subtract 3:
Term(n) = 2 - 3(n-1)
Combining these two cases into one expression, we can simplify it further:
Term(n) = 2 + (-1)^n * 3(n-1)
So, the general term in simplest form for the given sequence is:
Term(n) = 2 + (-1)^n * 3(n-1)