What is the Remainder of a Geometric Series when Divided by 5?

In this detailed guide, we will explore how to find the remainder of the sum of a geometric series when divided by 5. Specifically, we will consider the sum:

S 1 2 2^2 2^3 ... 2^{2015}

After discussing the general approach and providing step-by-step calculations, we will understand the concept of geometric series and modulo arithmetic.

Introduction to Geometric Series and Summation

A geometric series is a series with a constant ratio between successive terms. For instance, in our case, the series is:

1, 2, 4, 8, ..., 2^{2015}

The formula for the sum of a geometric series is given by:

S_n a frac{r^n - 1}{r - 1}

where a is the first term, r is the common ratio, and n is the number of terms. In this series, the first term a 1, the common ratio r 2, and there are 2016 terms from 0 to 2015.

Step-by-Step Calculation

First, we substitute the values into the formula:

S frac{2^{2016} - 1}{2 - 1} 2^{2016} - 1

Next, we need to find the remainder of this sum when divided by 5. This means we need to calculate:

2^{2016} - 1 mod 5

Understanding Modulo Arithmetic

To find the remainder, we first observe the pattern of the powers of 2 modulo 5:

2^0 equiv 1 mod 5 2^1 equiv 2 mod 5 2^2 equiv 4 mod 5 2^3 equiv 3 mod 5 2^4 equiv 1 mod 5

We see that the pattern repeats every 4 terms. So, to find 2^{2016} mod 5, we reduce the exponent modulo 4:

2016 mod 4 0

Thus:

2^{2016} equiv 2^0 equiv 1 mod 5

Final Calculation

Substituting this back into our expression for S, we get:

S equiv 2^{2016} - 1 equiv 1 - 1 equiv 0 mod 5

This shows that the sum S is perfectly divisible by 5, and the remainder is 0.

Alternative Method

We can also use the fact that:

x^n - 1 / (x - 1) 1 x x^2 ... x^n

For our specific case:

S 2^{2016} - 1

We know that:

2^2 equiv 4 equiv -1 mod 5

Therefore:

2^{2016} (2^2)^{1008} equiv (-1)^{1008} equiv 1 mod 5

And:

S equiv 2^{2016} - 1 equiv 1 - 1 equiv 0 mod 5

Again, we find that the remainder is 0.

Conclusion

In conclusion, the remainder when the sum S 1 2 2^2 ... 2^{2015} is divided by 5 is 0. This result is derived using both the direct formula for geometric series and the properties of modulo arithmetic.

Further Reading

If you find this topic interesting and want to explore more, consider delving into series and sequences, number theory, and modular arithmetic. These concepts are crucial in solving a wide range of problems and are widely used in various fields such as computer science, cryptography, and engineering.