What is the formula for the nth term of this sequence:
2, 6, 12, 20...
2 , 6 , 12 , 20...
n = 1
a1 = 2 = 1 + 1 = 1 ^ 2 + 1 = n ^ 2 + n
n = 2
a2 = 6 = 4 + 2 = 2 ^ 2 + 2 = n ^ 2 + n
n = 3
a3 = 12 = 9 + 3 = 3 ^ 2 + 3 = n ^ 2 + n
n = 4
a3 = 20 = 16 + 4 = 4 ^ 2 + 4 = n ^ 2 + n
an = n ^ 2 + n
what are the next 2 term of the arithmetic sequence below
6,12,18, , ,
To find the formula for the nth term of the given sequence, we will analyze the differences between consecutive terms.
The differences between consecutive terms are:
6 - 2 = 4
12 - 6 = 6
20 - 12 = 8
As we can see, the differences between the consecutive terms are increasing by 2 each time.
Therefore, we can conclude that the formula for the nth term of the sequence is given by:
nth term = (n^2) + (n + 1)
For example,
- When n = 1, the first term is 2: (1^2) + (1 + 1) = 2.
- When n = 2, the second term is 6: (2^2) + (2 + 1) = 6.
- When n = 3, the third term is 12: (3^2) + (3 + 1) = 12.
- When n = 4, the fourth term is 20: (4^2) + (4 + 1) = 20.
Using this formula, we can find any term in the sequence by substituting the value of n.
To find the formula for the nth term of a sequence, we need to look for a pattern or relationship between the terms. Let's analyze the given sequence: 2, 6, 12, 20...
Notice that each term can be obtained by multiplying a number by n and adding a constant term. In this case, when n is 1, the first term is 2. When n is 2, the second term is 6. When n is 3, the third term is 12. And when n is 4, the fourth term is 20.
Let's try multiplying n by a number and see if it matches the sequence. If we multiply n by 2, we get the following terms: 2, 4, 6, 8. These do not match the sequence.
Now, let's try multiplying n by 4, since it seems to be increasing by 4 each time. If we multiply n by 4, we get the following terms: 4, 8, 12, 16. These match the sequence, but they are not exactly the same. It seems like we need to add a constant term as well.
Subtracting the original sequence from the sequence obtained by multiplying n by 4, we get: 2, 2, 0, -4. This shows us that we need to add 2 to each term.
Now we have the correct formula: n * 4 + 2.
Therefore, the formula for the nth term of this sequence is 4n + 2.