find the expression in terms of n of the nth term of following squences
5,13,32,69,129,221....help plz show it work
I looked at that yesterday, but I could not find an obvious pattern
Found this:
5 = 2^2 + 1^2
13 = 3^2 + 2^2
but then it broke apart on the next one
Took differences and got:
5
13 -- 8
32 -- 19 -- 11
69 -- 37 -- 18 -- 7
129 - 60 -- 23 -- 5 -- 2
221 - 92 -- 32 -- 9 -- 4
not enough terms to tell if the 5th difference column is linear.
If the 6th column turns out to be a constant , then you would have a 6th degree polynomial
sorry, can't find out more at this time
I suspect a typo.
6 = 1^3+5
13 = 2^3+5
32 = 3^3+5
69 = 4^3+5
130 = 5^3+5
221 = 6^3+5
If no typos, then I'm stuck, too.
To find the expression for the nth term of a sequence, we need to look for a pattern in the sequence. Let's examine the differences between consecutive terms to see if there is a consistent pattern.
5, 13, 32, 69, 129, 221
The differences are:
13 - 5 = 8
32 - 13 = 19
69 - 32 = 37
129 - 69 = 60
221 - 129 = 92
We can observe that the differences between consecutive terms are increasing by consecutive odd numbers: 8, 19, 37, 60, 92.
In order to obtain the formula for the nth term, we need to generate a sequence that consists of the differences between consecutive terms. Let's generate this sequence:
Difference sequence: 8, 19, 37, 60, 92
Now, we need to examine the differences between the consecutive terms in the difference sequence:
19 - 8 = 11
37 - 19 = 18
60 - 37 = 23
92 - 60 = 32
Again, we notice that the differences between consecutive terms are increasing by consecutive even numbers: 11, 18, 23, 32.
To find the formula for the nth term, we need to generate a sequence that consists of the differences between consecutive terms in the difference sequence. Let's generate this sequence:
Difference of differences: 11, 18, 23, 32
Finally, we can observe that the differences of differences are increasing by consecutive odd numbers: 11, 18, 23, 32.
From this pattern, we can conclude that the expression for the nth term of the given sequence is a quadratic expression in terms of n.
To find the expression, we start with the quadratic form:
an² + bn + c
Now, we need to determine the values of a, b, and c.
First, let's consider the first term of the difference of differences sequence (11). This term corresponds to n = 3 (the third term of the original sequence). Plugging in n = 3, we get:
a(3²) + b(3) + c = 11
9a + 3b + c = 11 ------(Equation 1)
Next, let's consider the second term of the difference of differences sequence (18). This term corresponds to n = 4 (the fourth term of the original sequence). Plugging in n = 4, we get:
a(4²) + b(4) + c = 18
16a + 4b + c = 18 ------(Equation 2)
Lastly, let's consider the third term of the difference of differences sequence (23). This term corresponds to n = 5 (the fifth term of the original sequence). Plugging in n = 5, we get:
a(5²) + b(5) + c = 23
25a + 5b + c = 23 ------(Equation 3)
Now, we have a system of three equations (Equation 1, Equation 2, and Equation 3), which can be solved simultaneously to find the values of a, b, and c.
Subtracting Equation 1 from Equation 2, we eliminate c:
(16a + 4b + c) - (9a + 3b + c) = 18 - 11
16a - 9a + 4b - 3b = 7
7a + b = 7 ------(Equation 4)
Subtracting Equation 2 from Equation 3, we eliminate c:
(25a + 5b + c) - (16a + 4b + c) = 23 - 18
25a - 16a + 5b - 4b = 5
9a + b = 5 ------ (Equation 5)
Now, we have a system of two equations (Equation 4 and Equation 5) with two variables (a and b). Solving this system will give us the values of a and b.
Multiplying Equation 4 by 9 and Equation 5 by 7, we get:
63a + 9b = 63 -------(Equation 6)
63a + 7b = 35 -------(Equation 7)
Subtracting Equation 7 from Equation 6, we eliminate a:
(63a + 9b) - (63a + 7b) = 63 - 35
2b = 28
b = 14
Substituting b = 14 in Equation 4, we can solve for a:
7a + 14 = 7
7a = -7
a = -1
Now that we have the values of a and b, we can substitute them back into any of the three equations (Equation 1, Equation 2, or Equation 3) to solve for c.
Using Equation 1:
9a + 3b + c = 11
9(-1) + 3(14) + c = 11
-9 + 42 + c = 11
33 + c = 11
c = 11 - 33
c = -22
Therefore, the expression for the nth term of the given sequence is:
n² - 14n - 22