Find formula Number Sequences of
3, 10, 31, 94,.....
To find the formula for a number sequence, we need to observe the pattern or relationship between the numbers. Let's take a look at the given sequence: 3, 10, 31, 94, ...
From the first number (3) to the second number (10), we see that it increased by 7 (10 - 3 = 7). From the second number (10) to the third number (31), it increased by 21 (31 - 10 = 21). From the third number (31) to the fourth number (94), it increased by 63 (94 - 31 = 63).
If we observe closely, it seems that each number in the sequence is obtained by adding a multiple of the same number. Let's call this number "x". So, the second number is 3 + 1 * x, the third number is 3 + 2 * x, the fourth number is 3 + 3 * x, and so on.
Now, let's calculate the value of "x" by comparing the difference between the numbers:
Difference between the second and first numbers: 10 - 3 = 7
Difference between the third and second numbers: 31 - 10 = 21
Difference between the fourth and third numbers: 94 - 31 = 63
The differences are not consistent, indicating that the value of "x" is changing. To find the value of "x", we need to look for a pattern in the differences. Let's calculate the differences between the differences:
Difference between the second and first differences: 21 - 7 = 14
Difference between the third and second differences: 63 - 21 = 42
The second differences are consistent. They are increasing by 28 (42 - 14 = 28).
This implies that the "x" value is changing by the second differences, which means "x" itself is increasing by 28 each time.
Now, let's find the value of "x" for the first number to get the formula:
3 + 0 * 28 = 3
Therefore, the formula for the given number sequence is: 3 + (n - 1) * 28, where "n" represents the position of the number in the sequence.
Using this formula, we can find any term in the sequence. For example, for the 5th term, we substitute n = 5 into the formula:
3 + (5 - 1) * 28 = 3 + 4 * 28 = 3 + 112 = 115
So, the 5th term in the sequence is 115.
To find the formula for the number sequence, we first need to identify the pattern or rule that governs the sequence. By observing the differences between consecutive terms, we can determine the pattern:
10 - 3 = 7
31 - 10 = 21
94 - 31 = 63
The differences between consecutive terms (also called the first differences) are increasing by a constant amount, which is 7. Let's calculate the second differences:
21 - 7 = 14
63 - 21 = 42
The second differences are also increasing by a constant amount, which is 14. Therefore, we can conclude that the sequence is created by a quadratic equation.
To determine the quadratic equation, we'll use the general form: an^2 + bn + c, where n represents the position of the term in the sequence.
Let's substitute the values of n and the corresponding terms from the sequence into the equation to form a system of equations:
When n = 1, the corresponding term is 3:
a(1)^2 + b(1) + c = 3
a + b + c = 3 (Equation 1)
When n = 2, the corresponding term is 10:
a(2)^2 + b(2) + c = 10
4a + 2b + c = 10 (Equation 2)
When n = 3, the corresponding term is 31:
a(3)^2 + b(3) + c = 31
9a + 3b + c = 31 (Equation 3)
We now have a system of three equations:
Equation 1: a + b + c = 3
Equation 2: 4a + 2b + c = 10
Equation 3: 9a + 3b + c = 31
Solving this system of equations will give us the values of a, b, and c, allowing us to form the formula for the sequence.
not arithmetic nor geometric
gave up and checked with Wolfram
found:
term(n) = (7(3^n) - 3)/6
http://www.wolframalpha.com/input/?i=pattern+%7B3,+10,+31,+94%7D
quite obscure, perhaps more terms would have helped.