How do you get the general term for a sequence that has increasing common differences? For example: 1,2,4,7,11
notice that the differences are 1,2,3,4,...
but the differences of those differences are 1,1,1,...
Since we have constants in the second set of differences our sequence can be expressed as a 2nd degree expression
(had the third set of differences been constant, it would have been a cubic expression, etc)
so let the sequence be written as
t(n) = an^2 + bn + c
if n=1 ---> a + b + c = 1
if n=2 ---> 4a + 2b + c = 2
if n=3 ---> 9a + 3b + c = 4
subtract the first two ---> 3a + b = 1
subtract the last two ---> 5a + b = 2
now subtract those
2a = 1
a = 1/2
back in 3a+b=1
3/2 + b = 1
b = -1/2
back in a+b+c=1
1/2 - 1/2 + c = 1
c=1
so t(n) = (1/2)n^2 - (1/2)n + 1
= (1/2)(n^2 - n + 2)
checking for t(5)
= (1/2)(25-5+2)
=(1/2)(22) = 11
why do you let n=1, 2 and 3?
To get the general term for a sequence that has increasing common differences, you can use a method called finite differences.
1. Start by writing out the given sequence: 1, 2, 4, 7, 11.
2. Find the differences between consecutive terms and write them below the sequence:
1, 2, 4, 7, 11
1, 2, 3, 4
3. Repeat step 2 until you have a column of constant differences:
1, 2, 4, 7, 11
1, 2, 3, 4
1, 1, 1
4. Once you have a column of constant differences, you can use this column to determine the general term of the sequence.
In this case, the last column consists of all 1's, so this indicates that the original sequence has a second-order difference that is constant.
5. To find the general term for the sequence, we can use the formula for the nth term of a quadratic sequence:
T(n) = an^2 + bn + c, where a, b, and c are constants to be determined.
6. We know that the second-order difference is constant (1), so this means that the coefficient of the quadratic term (a) is 1/2.
7. To find the values of b and c, we can substitute some known values from the sequence into the general term formula. Let's use the first three terms: T(1) = 1, T(2) = 2, and T(3) = 4.
- Substituting T(1) = 1:
1 = a(1^2) + b(1) + c
- Substituting T(2) = 2:
2 = a(2^2) + b(2) + c
- Substituting T(3) = 4:
4 = a(3^2) + b(3) + c
8. Solve the system of three equations from step 7 to find the values of a, b, and c. In this case, the values are a = 1/2, b = 1/2, and c = 0.
9. Finally, substitute the values of a, b, and c into the general term formula to obtain the specific general term for the given sequence: T(n) = (1/2)n^2 + (1/2)n.
Therefore, the general term for the sequence 1, 2, 4, 7, 11 is T(n) = (1/2)n^2 + (1/2)n.