The first term of a sequence is 2014. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the `2014 ^(th)` term of the sequence ?

Answer

To find the 2014th term of the sequence where the first term is 2014 and each succeeding term is the sum of the cubes of the digits of the previous term, we can follow these steps:

### Step 1: Identify the first term
The first term of the sequence is given as:
\[ a_1 = 2014 \]

### Step 2: Calculate the second term
To find the second term \( a_2 \), we need to calculate the sum of the cubes of the digits of \( a_1 \):
- The digits of 2014 are 2, 0, 1, and 4.
- Calculate the cubes:
- \( 2^3 = 8 \)
- \( 0^3 = 0 \)
- \( 1^3 = 1 \)
- \( 4^3 = 64 \)

Now, sum these cubes:
\[ a_2 = 8 + 0 + 1 + 64 = 73 \]

### Step 3: Calculate the third term
Next, we find the third term \( a_3 \) by calculating the sum of the cubes of the digits of \( a_2 \):
- The digits of 73 are 7 and 3.
- Calculate the cubes:
- \( 7^3 = 343 \)
- \( 3^3 = 27 \)

Now, sum these cubes:
\[ a_3 = 343 + 27 = 370 \]

### Step 4: Calculate the fourth term
Now, we find the fourth term \( a_4 \) by calculating the sum of the cubes of the digits of \( a_3 \):
- The digits of 370 are 3, 7, and 0.
- Calculate the cubes:
- \( 3^3 = 27 \)
- \( 7^3 = 343 \)
- \( 0^3 = 0 \)

Now, sum these cubes:
\[ a_4 = 27 + 343 + 0 = 370 \]

### Step 5: Identify the pattern
Notice that \( a_4 = 370 \) and since \( a_3 \) is also 370, we can see that all subsequent terms will also be 370:
\[ a_5 = 370, \quad a_6 = 370, \quad \ldots \]

### Step 6: Conclusion
Since the sequence stabilizes at 370 from the third term onward, the 2014th term of the sequence is:
\[ a_{2014} = 370 \]

### Final Answer
Thus, the 2014th term of the sequence is:
\[ \boxed{370} \]

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