if the sequence is
2,15,52,125,246....
find the nth term expression
and sum of the nth terms
the growth does not appear exponential, yet it isn't just an easy power, either.
Note that the terms are
2*1, 3*5, 4*13, 5*25, 6*41, ...
The first factor is easy: (n+1)
The second factor is the sequence
1, 5, 13, 25, 41
1st difference: 4,8,12,16, ...
2nd difference: 4,4,4, ...
So, it is a quadratic. Some investigation shows that it is
(n-1)^2 + n^2 = 2n^2-2n+1
So, the nth term is
(n+1)(2n^2-2n+1) = 2n^3-n+1
Now, the sum of then 1st n terms will be a 4th-power polynomial.
So,
Sn = 2∑k^3 - ∑k + ∑1
k=1..n
You probably have handy the needed formulas.
To find the nth term expression of a sequence, we need to identify the pattern or rule that relates each term to its position in the sequence. Let's examine the given sequence to identify this pattern:
2, 15, 52, 125, 246
After observing the terms, it appears that a pattern is emerging. Let's analyze the differences between consecutive terms:
15 - 2 = 13
52 - 15 = 37
125 - 52 = 73
246 - 125 = 121
The differences between consecutive terms are not constant, but if we observe the differences of these differences, we see that they follow a consistent pattern:
37 - 13 = 24
73 - 37 = 36
121 - 73 = 48
Now, we can observe that the second differences are constant. This indicates that the nth term will be a quadratic expression.
To find the nth term expression, we need to represent the pattern in terms of an equation.
Let's denote the position of the term as n and the term itself as T(n).
From our analysis, we can conclude that the nth term expression can be written in the form:
T(n) = an^2 + bn + c
To determine the values of a, b, and c, we can use the differences we calculated earlier.
First differences:
T(2) - T(1) = 15 - 2 = 13
T(3) - T(2) = 52 - 15 = 37
T(4) - T(3) = 125 - 52 = 73
T(5) - T(4) = 246 - 125 = 121
Second differences:
37 - 13 = 24
73 - 37 = 36
121 - 73 = 48
The second differences are constant, so we can assign the value of 24 to 2a.
24 = 2a
a = 12
Now, substitute the value of a into any of the earlier first differences to solve for b:
13 = 12(2) + b
13 = 24 + b
b = -11
Finally, substituting the values of a and b into the equation and solving for c:
T(1) = a(1)^2 + b(1) + c
2 = 12(1)^2 + (-11)(1) + c
2 = 12 - 11 + c
2 = 1 + c
c = 1
Therefore, the nth term expression for the given sequence is:
T(n) = 12n^2 - 11n + 1
Now, let's find the sum of the nth terms.
The sum of the first n terms of a sequence can be found using the formula:
Sum(n) = (n/2) * (T(1) + T(n))
Substituting the values into the formula:
Sum(n) = (n/2) * (2 + T(n))
= (n/2) * (2 + 12n^2 - 11n + 1)
= (n/2) * (12n^2 - 9n + 3)
So, the sum of the nth terms can be calculated using the expression:
Sum(n) = (n/2) * (12n^2 - 9n + 3)