Let `f(x) = a_0+a_1x+a_2x^2+........a_nx^n `, where `a_i` are non-negative integers for `i = 0, 1, 2..........n`. If `f(1) = 21` and `f(25) = 78357`. Then, find the value of `f(10)`.

Explanation

To solve the problem step by step, we start with the polynomial function given by:

\[ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \ldots + a_n x^n \]

where \( a_i \) are non-negative integers.

### Step 1: Set up the equations using the given values

We know:
1. \( f(1) = 21 \)
2. \( f(25) = 78357 \)

From \( f(1) = 21 \), we can write:

\[ a_0 + a_1 + a_2 + a_3 + \ldots + a_n = 21 \quad \text{(Equation 1)} \]

From \( f(25) = 78357 \), we can write:

\[ a_0 + 25a_1 + 625a_2 + 15625a_3 + 390625a_4 + \ldots = 78357 \quad \text{(Equation 2)} \]

### Step 2: Analyze the degree of the polynomial

Since \( 25^4 = 390625 \) is greater than \( 78357 \), we conclude that \( a_4, a_5, \ldots, a_n \) must be zero. Thus, we only need to consider \( a_0, a_1, a_2, a_3 \).

### Step 3: Rewrite Equation 2

Now, we can simplify Equation 2 to:

\[ a_0 + 25a_1 + 625a_2 + 15625a_3 = 78357 \]

### Step 4: Substitute \( a_0 \) from Equation 1 into Equation 2

From Equation 1, we can express \( a_0 \) as:

\[ a_0 = 21 - a_1 - a_2 - a_3 \]

Substituting this into Equation 2 gives:

\[ (21 - a_1 - a_2 - a_3) + 25a_1 + 625a_2 + 15625a_3 = 78357 \]

### Step 5: Simplify the equation

This simplifies to:

\[ 21 + 24a_1 + 624a_2 + 15624a_3 = 78357 \]

Subtracting 21 from both sides:

\[ 24a_1 + 624a_2 + 15624a_3 = 78336 \]

### Step 6: Divide the entire equation by 24

Dividing through by 24 gives:

\[ a_1 + 26a_2 + 651a_3 = 3264 \quad \text{(Equation 3)} \]

### Step 7: Solve for \( a_3 \)

Since \( a_3 \) must be a non-negative integer, we can try different values for \( a_3 \) to find suitable values for \( a_1 \) and \( a_2 \).

1. **If \( a_3 = 5 \)**:
\[ a_1 + 26a_2 + 651 \cdot 5 = 3264 \\ a_1 + 26a_2 + 3255 = 3264 \\ a_1 + 26a_2 = 9 \]
This gives us possible values for \( a_1 \) and \( a_2 \):
- If \( a_2 = 0 \), then \( a_1 = 9 \).
- If \( a_2 = 1 \), then \( a_1 = -17 \) (not valid).
- Therefore, \( a_2 = 0 \) and \( a_1 = 9 \).

### Step 8: Find \( a_0 \)

Using \( a_0 + a_1 + a_2 + a_3 = 21 \):

\[ a_0 + 9 + 0 + 5 = 21 \]

Thus,

\[ a_0 = 21 - 14 = 7 \]

### Step 9: Values of coefficients

We have found:
- \( a_0 = 7 \)
- \( a_1 = 9 \)
- \( a_2 = 0 \)
- \( a_3 = 5 \)

### Step 10: Write the polynomial

Now we can write the polynomial:

\[ f(x) = 7 + 9x + 5x^3 \]

### Step 11: Calculate \( f(10) \)

Now we can find \( f(10) \):

\[ f(10) = 7 + 9(10) + 5(10^3) \\ = 7 + 90 + 5000 \\ = 5097 \]

### Final Answer

Thus, the value of \( f(10) \) is:

\[ \boxed{5097} \]

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