Python Program to calculate n+nm+nmm.......+n(m times).

Here we need to calculate a series where each term is formed by repeating digit n multiple times: n + nn + nnn + ... up to m terms. For example, if n=3 and m=4, the series would be 3 + 33 + 333 + 3333.

Algorithm

Step 1: Input n and m values
Step 2: Convert n to string for concatenation
Step 3: Initialize sum with first term (n)
Step 4: Build each term by concatenating n repeatedly
Step 5: Convert string back to integer and add to sum
Step 6: Return final sum

Example

Let's implement the solution to calculate the series sum ?

def sum_of_series(n, m):
    str_n = str(n)
    total_sum = n
    current_term = str_n
    
    for i in range(1, m):
        current_term = current_term + str_n
        total_sum = total_sum + int(current_term)
    
    return total_sum

# Example usage
n = 3
m = 5
result = sum_of_series(n, m)
print(f"For n={n} and m={m}")
print(f"Series: 3 + 33 + 333 + 3333 + 33333")
print(f"Sum of series: {result}")

For n=3 and m=5
Series: 3 + 33 + 333 + 3333 + 33333
Sum of series: 37035

Step-by-Step Breakdown

Let's trace through the example with n=3 and m=5 ?

def sum_of_series_detailed(n, m):
    str_n = str(n)
    total_sum = n
    current_term = str_n
    
    print(f"Term 1: {n}")
    
    for i in range(1, m):
        current_term = current_term + str_n
        term_value = int(current_term)
        total_sum = total_sum + term_value
        print(f"Term {i+1}: {term_value}")
    
    return total_sum

n = 3
m = 5
result = sum_of_series_detailed(n, m)
print(f"\nTotal sum: {result}")

Term 1: 3
Term 2: 33
Term 3: 333
Term 4: 3333
Term 5: 33333

Total sum: 37035

Alternative Mathematical Approach

We can also solve this using a mathematical formula without string concatenation ?

def sum_of_series_math(n, m):
    total_sum = 0
    
    for i in range(1, m + 1):
        # Calculate n repeated i times: n * (10^i - 1) / 9
        term = n * (10**i - 1) // 9
        total_sum += term
        
    return total_sum

# Test with same values
n = 3
m = 5
result = sum_of_series_math(n, m)
print(f"Using mathematical approach: {result}")

Using mathematical approach: 37035

Comparison

Conclusion

The string concatenation method is intuitive and easy to understand, while the mathematical approach using the formula n×(10^i?1)/9 is more efficient. Both methods correctly calculate the series n + nn + nnn + ... for m terms.