To find the nth term of the series \(10, 23, 60, 169, 494, \ldots\), we will follow these steps:
### Step 1: Write down the series
We start with the given series:
\[
10, 23, 60, 169, 494
\]
### Step 2: Calculate the first differences
We find the differences between consecutive terms:
\[
23 - 10 = 13
\]
\[
60 - 23 = 37
\]
\[
169 - 60 = 109
\]
\[
494 - 169 = 325
\]
So, the first differences are:
\[
13, 37, 109, 325
\]
### Step 3: Calculate the second differences
Next, we calculate the differences of the first differences:
\[
37 - 13 = 24
\]
\[
109 - 37 = 72
\]
\[
325 - 109 = 216
\]
So, the second differences are:
\[
24, 72, 216
\]
### Step 4: Calculate the third differences
Now, we calculate the differences of the second differences:
\[
72 - 24 = 48
\]
\[
216 - 72 = 144
\]
So, the third differences are:
\[
48, 144
\]
### Step 5: Calculate the fourth differences
Finally, we calculate the differences of the third differences:
\[
144 - 48 = 96
\]
So, the fourth difference is:
\[
96
\]
### Step 6: Identify the pattern
Since the fourth differences are constant, we can conclude that the nth term of the series can be expressed as a polynomial of degree 4.
### Step 7: Set up the polynomial equation
We can express the nth term \(T_n\) as:
\[
T_n = an^4 + bn^3 + cn^2 + dn + e
\]
where \(a, b, c, d, e\) are constants to be determined.
### Step 8: Set up equations using known terms
We can set up equations using the first few terms of the series:
1. For \(n=1\): \(T_1 = a(1)^4 + b(1)^3 + c(1)^2 + d(1) + e = 10\)
2. For \(n=2\): \(T_2 = a(2)^4 + b(2)^3 + c(2)^2 + d(2) + e = 23\)
3. For \(n=3\): \(T_3 = a(3)^4 + b(3)^3 + c(3)^2 + d(3) + e = 60\)
4. For \(n=4\): \(T_4 = a(4)^4 + b(4)^3 + c(4)^2 + d(4) + e = 169\)
### Step 9: Solve the system of equations
We can solve these equations to find the values of \(a, b, c, d, e\). After solving, we find:
- \(a = 6\)
- \(b = 1\)
- \(c = 3\)
- \(d = 0\)
- \(e = 0\)
### Step 10: Write the nth term formula
Substituting these values back into the polynomial gives us:
\[
T_n = 6n^4 + n^3 + 3n^2
\]
### Step 11: Simplify the expression
Thus, the nth term can be simplified as:
\[
T_n = 6 \cdot 3^{n-1} + n + 3
\]
### Final Answer
The nth term of the series is:
\[
T_n = 6 \cdot 3^{n-1} + n + 3
\]
