sandra wrote the sequence below
2,5,10,17,...
which equation represents the rule for finding the nth term of this equation
did you notice that each term is one larger than a perfect square, ....
2 = 1^2 + 1
5 = 2^2 + 1
10 = 3^2 + 1
17 = 4^2 + 1
.
.
so what do you think?
The numbers are increasing by 3 then 5 then 7 then 9. Next, notice that every number is actually one more than a square number.
So the general formula is:
a(n)=1+n^2
Let's check:
a(1)=1+1^2=1+1=2
a(2)=1+2^2=1+4=5
a(3)=1+3^2=1+9=10
a(4)=1+4^2=1+16=17
We see it's correct.
The next numbers in the sequence should be:
a(5)=1+5^2=1+25=26
a(6)=1+6^2=1+36=37 etc.
To find the rule for the nth term of the given sequence, we need to observe the pattern between the numbers. Let's write down the differences between consecutive terms:
2 -> 5: 5 - 2 = 3
5 -> 10: 10 - 5 = 5
10 -> 17: 17 - 10 = 7
We can see that the differences are increasing by 2 each time. This means that the pattern for the differences is an arithmetic sequence with a common difference of 2.
To find the rule for the nth term, we can use the formula for arithmetic sequences:
nth term = first term + (n - 1) * common difference
In this case, the first term is 2 and the common difference is increasing by 2 each time. Therefore, the equation representing the rule for finding the nth term of the sequence is:
nth term = 2 + (n - 1) * 2
Simplifying this equation gives us:
nth term = 2 + 2n - 2
nth term = 2n
To find the equation that represents the rule for finding the nth term of the given sequence, we need to look for the pattern within the sequence.
Looking at the sequence:
2, 5, 10, 17, ...
We can start by finding the differences between consecutive terms:
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
The differences between consecutive terms seem to be increasing by 2 each time. This suggests that the sequence might be quadratic.
To verify this, let's find the differences between the differences:
5 - 3 = 2
7 - 5 = 2
The differences between the differences are the same, which indicates a quadratic pattern.
Since the differences between consecutive terms increase linearly, we can express the nth term of the sequence using a quadratic equation of the form:
an^2 + bn + c
To find the values of a, b, and c, we can substitute values from the sequence into the equation and solve the resulting system of equations.
Let's substitute the first few terms of the sequence:
When n = 1, a(1)^2 + b(1) + c = 2
When n = 2, a(2)^2 + b(2) + c = 5
When n = 3, a(3)^2 + b(3) + c = 10
Simplifying these equations, we have:
a + b + c = 2 -- Equation 1
4a + 2b + c = 5 -- Equation 2
9a + 3b + c = 10 -- Equation 3
Solving this system of equations, we find that a = 1, b = 1, and c = 0.
Therefore, the equation representing the rule for finding the nth term of the sequence is:
n^2 + n
So, the answer is (n^2 + n).