pattern 1,5,11,19 1.how do i find the next two patterns 2.how do i calculate the n^th term of the pattern
form 4 columns
x, y, first difference in y's, 2nd difference in y's
1 1
2 5 4
3 11 6 2
4 19 8 2
since the 2nd differences are constant, we have a quadratic
let the relation be
term(n) = an^2 + bn + c
for (1,1) --- 1 = a + b + c
for (2,5) --- 5 = 4a + 2b + c
for (3,11) --- 11 = 9a + 3b + c
subtract the first two:
3a + b = 4
subtract the 2nd from the 3rd:
5a + b = 6
subtract those last two:
2a = 2
a = 1
sub into 3a+b = 4 ---> b = 1
sub into a+b+c = 1
1 + 1 + c = 1
c = -1
rule: term(n) = n^2 + n - 1
my first 3 values are satisfied by this equation, let's check if the last one works
if x = 4
term(4) = 16 + 4 - 1 = 19
yeahhh!!!!
check the differences: 4,6,8
I suspect that the next terms are
19+10=29 and 29+12 = 41
you plug in 3 pairs of values into a general quadratic y=ax^2+bx+c and you get
a+b+c=1
4a+2b+c=5
9a+3b+c=11
a=1, b=1, c=-1
and get Tn = f(n) = x^2+x-1
Or, you can recall that
1+2+3+...n = n(n+1)/2
You have
1 + 4+6+8+...
= -1 + 2+4+6+...
= -1+2(1+2+3+...)
Tn = -1+2*n(n+1)/2
= -1 + n(n+1)
= n^2+n-1
thanks which term of the pattern
is equal to 2549
come on. You have the formula. Just solve
n^2+n-1 = 2549
Hint: it's not -51
To find the next two numbers in the pattern (1, 5, 11, 19), we need to identify the pattern or sequence that governs the numbers. By examining the differences between consecutive terms, we can determine if there is a consistent rule.
1. Finding the next two patterns:
To identify the pattern, let's calculate the differences between consecutive terms:
5-1 = 4
11-5 = 6
19-11 = 8
We notice that the differences (4, 6, 8) themselves form a pattern. This indicates that the original sequence may be quadratic or polynomial.
To find the next difference, we can extend the pattern:
4, 6, 8, 10, 12, 14, ...
Now, let's add the differences to the original sequence:
1 + 4 = 5
5 + 6 = 11
11 + 8 = 19
19 + 10 = 29
29 + 12 = 41
Thus, the next two numbers in the pattern are 29 and 41.
2. Calculating the nth term of the pattern:
To calculate the nth term of the pattern, we need to determine the formula or equation that generates the sequence. In this case, we suspect it might be a quadratic sequence.
By examining the pattern again, we can find the second difference of the differences:
6-4 = 2
8-6 = 2
The second differences are constant, indicating a quadratic sequence. We can proceed with the general formula for a quadratic sequence: an^2 + bn + c.
To find the coefficients a, b, and c, we can substitute the values from the pattern into the formula and solve the resulting system of equations. Since we have three points from the pattern (1, 5), (2, 11), and (3, 19), we can form the following equations:
a(1^2) + b(1) + c = 1
a(2^2) + b(2) + c = 5
a(3^2) + b(3) + c = 19
By simplifying and solving this system of equations, we can determine the coefficients a, b, and c. Once we have those, we can substitute any value of n to find the respective term in the sequence.
Note: Without further information, it is not possible to determine the nth term of the pattern definitively. However, if we assume a quadratic sequence based on the given terms, the above steps can be followed to calculate the potential nth term.