# Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two

Chinnaraji Annamalai

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Chinnaraji Annamalai. Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two. 2022. hal-03710654

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# Computation for the Summation of Binomial Expansions and Geometric Series of Multiples of Powers of Two

Chinnaraji Annamalai

School of Management, Indian Institute of Technology, Kharagpur, India

Email: anna@iitkgp.ac.in

https://orcid.org/0000-0002-0992-2584

**Abstract:** This paper presents computing technique for the summation of binomial expansions and geometric series of multiples of powers of two. This computing technique is a methodological advance which is useful for researchers who are working in science, economics, engineering, computation, and management.

**MSC Classification codes:** 05A10, 40A05 (65B10)

**Keywords:** computation, combinatorics, geometric series, binomial coefficient

## 1. Introduction

The author of this article introduces an innovative binomial coefficient [1-3] along with a traditional binomial coefficient. The innovative binomial coefficient is shown below:

$$V_r^n = \frac{(r+1)(r+2)(r+3)\cdots(r+n)}{n!}, \quad (n \ge 1, r \ge 0 \ \& \ n, r \in N = \{0,1,2,3,\cdots\}).$$

Traditional binomial coefficient: $\binom{n}{r} = nCr = \frac{n!}{r!(n-r)!}$, where $n, r \in N$.

Let us compare the innovative binomial coefficient $V_x^y$ with the traditional binomial coefficient as follows:

Let $z = x + y$. Then, $zC_x = \frac{z!}{x!y!}$. Here, $V_x^y = V_y^x \Rightarrow zC_x = zC_y$, $(x, y, z \in N)$.

For example,

$$V_3^5 = V_5^3 = (5+3)C_3 = (5+3)C_5 = 56.$$

Also, $V_n^0 = V_0^n = nC_0 = nC_n = \frac{n!}{n!0!} = 1$ and $V_0^0 = 0C_0 = \frac{0!}{0!} = 1$ ($\because 0! = 1$).

## 2. Computation of Binomial Expansions and Geometric Series

In this section, the author of this article provides a theorem based on binomial expansions and geometric series [4-13].

**Theorem:** $\sum_{i=1}^{1} i \times V_i^{1-i} + \sum_{i=1}^{2} i \times V_i^{1-i} + \sum_{i=1}^{3} i \times V_i^{1-i} + \cdots + \sum_{i=1}^{n} i \times V_i^{1-i} = (n-1)2^n + 1$.

*Proof.* Let us find the value of each binomial expansion in the binomial theorem step by step.

**Step 1:** $1 \times V_1^0 = 1 \Rightarrow \sum_{i=1}^{1} i \times V_i^{1-i} = 1 = 1 \times 2^0$, ($\because V_n^0 = V_0^n = 1 \ \& \ n = 0, 1, 2, 3, \cdots$).
$$ \text{Step 2: } \sum_{i=1}^{2} i \times V_i^{1-i} = 1 \times V_1^1 + 2 \times V_2^0 = 2 + 2 = 4 = 2 \times 2^1. $$

$$ \text{Step 3: } \sum_{i=1}^{3} i \times V_i^{1-i} = 1 \times V_1^2 + 2 \times V_2^1 + 3 \times V_3^0 = 3 + 6 + 3 = 12 = 3 \times 2^2. $$

$$ \text{Step 4: } \sum_{i=1}^{4} i \times V_i^{1-i} = 1 \times V_1^3 + 2 \times V_2^2 + 3 \times V_3^1 + 4 \times V_4^0 = 4 + 12 + 12 + 4 = 4 \times 2^3. $$

Similarly, we can continue the expressions up to "step n" such that $\sum_{i=0}^{n} i \times V_i^{n-i} = n2^{n-1}$.

Now, by adding these expressions on both sides, it appears as follows:

$$ \sum_{i=1}^{1} i \times V_i^{1-i} + \sum_{i=1}^{2} i \times V_i^{1-i} + \sum_{i=1}^{3} i \times V_i^{1-i} + \sum_{i=1}^{4} i \times V_i^{1-i} + \dots + \sum_{i=1}^{n} i \times V_i^{1-i} = \sum_{i=1}^{n} i \times 2^{i-1}. $$

where $\sum_{i=1}^{n} i \times 2^i = (n-1)2^n + 1$ is the geometric series of multiples of powers of two[3].

$$ \therefore \sum_{i=1}^{1} i \times V_i^{1-i} + \sum_{i=1}^{2} i \times V_i^{1-i} + \sum_{i=1}^{3} i \times V_i^{1-i} + \dots + \sum_{i=1}^{n} i \times V_i^{1-i} = (n-1)2^n + 1. $$

Hence, theorem is proved.

### 3. Conclusion

This article presented a numerical computational technique for the summation of binomial expansions and geometric series of multiples of powers of two. This technique and its results can be useful for researchers who are working in science, economics, engineering, management, and medicine [14].

### References

[1] Annamalai, C. (2009) Annamalai's Binomial Identity and Theorem. SSRN4097907. http://dx.doi.org/10.2139/ssrn.4097907.

[2] Annamalai, C. (2009) My New Idea for Optimized Combinatorial Techniques. SSRN4078523. http://dx.doi.org/10.2139/ssrn.4078523.

[3] Annamalai, C. (2022) Computation for the Summation of Integers and Geometric Progression of Powers of Two. OSF preprints. https://www.doi.org/10.31219/osf.io/y35ma.

[4] Annamalai, C. (2009) A novel computational technique for the geometric progression of powers of two. Journal of Scientific and Mathematical Research, Vol.3, pp 16-17. https://doi.org/10.5281/zenodo.6642923.
[5] Annamalai, C. (2019) Extension of ACM for Computing the Geometric Progression. *Journal of Advances in Mathematics and Computer Science*, Vol. 31(5), pp 1-3.
https://doi.org/10.9734/JAMCS/2019/v31i530125.

[6] Annamalai, C. (2015) A Novel Approach to ACM-Geometric Progression. *Journal of Basic and Applied Research International*, Vol.2(1), pp 39-40.
https://www.ikppress.org/index.php/JOBARI/article/view/2946.

[7] Annamalai, C. (2017) Computational Modelling for the Formation of Geometric Series using Annamalai Computing Method. *Jñānābha*. Vol.47(2), pp 327-330.
https://zbmath.org/?q=an%3A1391.65005.

[8] Annamalai, C. (2022) Summations of Single Terms and Successive Terms of Geometric Series. SSRN4085922. http://dx.doi.org/10.2139/ssrn.4085922.

[9] Annamalai, C. (2022) Sum of Geometric Series with Negative Exponents. SSRN4088497.
http://dx.doi.org/10.2139/ssrn.4088497.

[10] Annamalai, C. (2022) Computation of Geometric Series in Different Ways. OSF Preprints. https://doi.org/10.31219/osf.io/kx7d8.

[11] Annamalai, C. (2022) Numerical Computational Method for Computation of Binomial Expansions and Geometric Series. *Cambridge Open Engage*.
https://www.doi.org/10.33774/coe-2022-pnx53-v7.

[12] Annamalai, C. (2022) Algorithmic and Numerical Techniques for Computation of Binomial and Geometric Series. *Cambridge Open Engage*.
https://www.doi.org/10.33774/coe-2022-pnx53-v8.

[13] Annamalai, C. (2009) A Binomial Expansion equal to Multiple of 2 with Non-Negative Exponents. SSRN4116671. http://dx.doi.org/10.2139/ssrn.4116671.

[14] Annamalai, C. (2010) Application of Exponential Decay and Geometric Series in Effective Medicine. *Advances in Bioscience and Biotechnology*, Vol. 1(1), pp 51-54.
https://doi.org/10.4236/abb.2010.11008.