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Balance is an attribute of life that is not only important but also a necessity to lead a healthy and successful life. Always inspired by the IB learner profile, I have realized that balance is such a criterion that is not only important for a healthy lifestyle, but also in several physical events that are observed in daily life. Being an inquirer, I have always tried to knock on every door of the event with the intrigued curiosity and eagerness of learning that I have observed in daily life that are relevant to my curriculum to understand the events more coherently and efficiently.

Series and Sequence of Topic – 1 of IB Mathematics have two sections – Arithmetic Sequence and Geometric Sequence. While working on this particular chapter, I realized that there is a lack of coordination between the two sections as we were taught the use of Arithmetic Sequence and Geometric Sequence separately. One of the most important attributes which were lacking in this was Balance between the two. We live in a dynamic world where nothing operates independently. This has triggered my curiosity. Is there any event that operates as arithmetic and a geometric operation simultaneously? How will the formulation of each sequence, individually, affect the other? How will there be a change in the parametric equation of each sequence if they work simultaneously in any life event?

To decipher the answers to the above questions, I started my research on it. Recently, as an initiative to prevent Global Warming, an NGO has started planting a fixed number of trees every year in a forest keeping in mind that every year there is a decrease in the number of plants by a relatively constant proportion. There, I have got the case study of exploring and establishing a balance between the two different sequences if they are prevalent simultaneously. To establish a formulation of a sequence where Arithmetic and Geometric Sequence work simultaneously, I have gone through a few research journals on Sequences and Series on ‘Discrete Mathematics by Elsevier’ and ‘Australian Mathematical Society by Cambridge Press’ and also gone through a few video lectures on Sequences and Series in Higher Mathematics in YouTube to achieve an in-depth overview of the subject. In this process, I have understood and learnt several known parameters and factors affecting each sequence independently but could not find any relationship between the two if they are acting simultaneously. This has led me to land on the research question.

The main motive of this exploration is to find the estimated number of trees after 5 years and hence, determination of a generic formula of number of trees in n^{th} year (n^{th} term of a sequence) if the growth rate in the number of trees of the forest is an arithmetic progressive function and the death rate is a geometric progressive function.

A sequence of number in which the next term is incremented or decremented by a constant value with respect to the present term is known as an Arithmetic Sequence. The value by which each term is being incremented or decremented with respect to the previous term is known as Common Difference of an Arithmetic Progression or an Arithmetic Sequence. The expression of n^{th} term of an Arithmetic Sequence and the sum of first *n* terms of the sequence are shown below:

*T _{n}* =

* *\(S_n =\frac{n}{2}[2a + (n - 1) d]\)

where,

*T _{n}* =

S_{n} = sum of first n terms of the sequence

a = first term of the sequence

d = common difference of the sequence

n = number of ter

A sequence of number in which the next term is the product of a constant ratio and the present term is known as Geometric Sequence. The number by which each term is multiplied to is known as Common Ratio of Geometric Progression or a Geometric Sequence. The expression of n^{th} term of a Geometric Sequence and the sum of first n terms of the sequence and the sum of infinite terms of the sequence are shown below:

*T _{n}* =

^{ }^{\(Sn = a\biggl(\frac{1\,-\,r\,^n}{1\,-\,r}\biggl)... if 0 < r < 1\)}

^{\(Sn = a \biggl(\frac{r\,^n\,-\,1}{r\,-\,1}\biggl)... if |r| > 1\)}

^{ }^{\(S ∞ =\frac{a}{1\,-\,r}... 0 < r < 1\)}

where,

*T _{n}* =

*S _{n}* = sum of first n terms of the sequence

*S*_{ ∞ }= sum of infinite terms of the sequence

*a *= first term of the sequence

*r* = common ratio of the sequence

*n* = number of term

In the initiative of afforestation by an NGO, the recordings were published in the book. The required parameters of this exploration are mentioned below.

The average number of plants in the garden was 8200 at the beginning of 2020. Due to several environmental factors, 5% of the plants died every year. After the initiative was taken, 600 plants were planted every year.

For the year 2021 -

∴ The number of trees before plantation

\(= 8200 -\frac{5}{100}× 8200 = 8200 - 410 = 7790\)

∴ The number of trees after plantation

= 7790 + 600 = 8390

For the year 2022 -

∴ The number of trees before plantation

\(= 8390 -\frac{5}{100}× 8390 = 7970. 5\)

∴ The number of trees after plantation

= 7970. 5 + 600 = 8570. 5

For the year 2023 -

∴ The number of trees before plantation

\(= 8570. 5 - \frac{5}{100}× 8570. 5\)

∴The number of trees after plantation

= 8141. 975 + 600 = 8741. 975

For the year 2024 -

∴ The number of trees before plantation

\(= 8741.975 - \frac{5}{100}× 8741. 975 = 8304. 87625\)

∴ The number of trees after plantation

= 8304. 87625 + 600 = 8904. 87625

For the year 2025 -

∴ The number of trees before plantation

\(= 8904. 87625 -\frac{5}{100}× 8904. 87625 ≈ 8459.63\)

∴The number of trees after plantation

= 8459. 63 + 600 ≈ 9059. 63

**Graphical Analysis of Plant count every year**

Year | Number of trees (approximated) |
---|---|

2020 | 8200 |

2021 | 8390 |

2022 | 8570 |

2023 | 8742 |

2024 | 8905 |

2025 | 9060 |

In the above-mentioned graph, the variation in the number of plants (estimated) is plotted with respect to the year (from 2020 to 2025). The years (considered to be the independent variable), is plotted along the X – Axis, and the number of plants (considered to be the dependent variable), is plotted along the Y – Axis. In the graph, when moved forward with time from 2020 to 2025, the number of trees has increased from 8200 to 9060. Hence, an increasing and linear relationship has been obtained between the plant count and the number of years. The equation of the obtained trend between the considered dependent variable and the independent variable is: y = 171.91*x* - 339052 where *y* represents the number of plants and *x* represents years. It is clearly observed in the graph that all the data points are lying on the trendline with a very minute positive deviation of only one data point (*x* = 2023). Due to absence of any significant data point, the obtained relationship can be assumed to be stable. It is also verified by such a high value of the regression correlation coefficient of 0.99.

From the above graphical analysis, it is clear that, in any event which operates on both arithmetic progression as well as geometric progression, simultaneously, varies linearly.

The above discussion (refer to section 4.2), comprise analysis and equation of the trend specific to the particular event, i.e., according to the collected data. If the growth rate or death rate would have been changed, then the equation of the trend would no longer be valid. However, the estimated number of tree (*n*^{th} term of the sequence) that would be present in any specific year could be determined using the equation of the trend as shown in section 4.2. Thus, to establish a formula of the nth term of the sequence that would be valid for any value of common ratio and any value of common difference, the following exploration was done. It should be noted that, the following exploration is based on the same data set.

Let, the number of trees in year 2020, 2021, 2022, 2023, 2024, 2025 be Z_{0}, Z_{1}, Z_{2}, Z_{3}, Z_{4}, Z_{5}, and Z_{6}. Similar to the calculation shown above in section 4.2, here, number of trees in different years will be found in a different methodology.

For year 2020 -

*Z*_{0 }= 8200

For year 2021-

*Z*_{1 }= *Z*_{0} × 0.95 + 600

*Z*_{1} = 8200 × 0.95 + 600

For year 2022 -

*Z*_{2 }= *Z*_{1 }× 0.95 + 600

*Z*_{2 }= (8200 × 0.95 + 600) × 0.95 + 600

*Z*_{2 }= 8200 × 0.95^{2} + 600 × 0.95 + 600

*Z*_{2 }= 8200 × 0.95^{2} + 600 (0.95 + 1)

For year 2023 -

*Z*_{3 }= *Z*_{2} × 0. 95 + 600

\(Z3 =\biggl[8200 × 0. 952 + 600 (0. 95 + 1)× 0. 95 + 600\biggl{]}\)\(\)

*Z*_{3 }= 8200 × 0.95^{3} + 600 (0.95^{2} + 0.95) + 600

*Z*_{3 }= 8200 × 0. 95^{3} + 600 (0.95^{2} + 0.95 + 1)

For year 2024 -

*Z*_{4 }= *Z*_{3} × 0.95 + 600

\(Z4 =\biggl[8200 × 0. 953 + 600 (0. 952 + 0.95 + 1)\biggl] × 0. 95 + 600 \)

*Z*_{4 }= 8200 × 0.95^{4} + 600 (0.95^{3} + 0.95^{2} + 0.95) + 600

*Z*_{4 }= 8200 × 0.95^{4} + 600 (0.95^{3} + 0.95^{2} + 0.95 + 1)

For year 2025 -

Z_{5 }= Z_{4} × 0.95 + 600

\( Z5 = \biggl[8200 × 0. 954 + 600 (0. 953 + 0. 952 + 0. 95 + 1)\biggl] × 0. 95 + 600 \)

Z_{5 }= 8200 × 0.95^{5} + 600 (0.95^{4} + 0.95^{3} + 0.95^{2} + 0. 95) + 600

Z_{5 }= 8200 × 0.95^{4} + 600 (0.95^{4} + 0.95^{3} + 0.95^{2} + 0.95 + 1)

From the above calculation, a generalized formula of number of plants in n^{th} year could be expressed as -

Z_{n }= 8200 × 0.95^{n }+ 600 (0.95^{n - 1} + 0.95^{n-2 }+ ... + 0.95 + 1)………(equation-1)

If the common ratio, and the common difference are assumed to be *r* and *d* respectively, then equation (1) could be written in a general way like -

\(Zn = Z0 × (1 - r)n + d\biggl[(1 - r)n-1 + (1 - r)n-2 + … + (1 - r) + (1 - r)0\biggl] … (equation-2) \)

**Calculation of the sum of exponents**

(1 - r)^{n - 1} + (1 - r)^{n - 2} + ... + (1 - r) + (1 - r)^{0}

= (1 - r)^{0} + (1 - r) + ... + (1 - r)^{n - 2} + (1 - r)^{n - 1}

= 1 + (1 - r) + ... + (1 - r)^{n - 2} +(1 - r)^{n-1}

^{\(=\sum\limits^{n}_{n=1}1×(1 - r)^{n-1}\)}

Applying the formula of sum of *n* terms of a geometric progression (refer to section 3.2):

^{ }^{\(\sum\limits^{n}_{n=1}1×(1 - r)^{n-1}\)}

^{\(=1\biggl(\frac{1-(1-r)^n}{1-(1-r)}\biggl)\)}

\(=\frac{1-(1-r)^n}{r}\)

From equation (2) -

\(Zn = Z0 × (1 - r)n + d\biggl[ (1 - r)n - 1 + (1 - r)n - 2 + ... + (1 - r) + (1 - r)0\biggl]\)\(\)

\(Zn = Z0 × (1 - r)n + d \biggl[\frac{1-(1-r)^{n}}{r}\biggl]\)

\(Zn = Z0 × (1 - r)n - d \biggl[\frac{(1-r)^{n}-1^{}}{r}\biggl]\)

\(Zn = Z0 × (1 - r)n - d\biggl[\frac{(1-r)^{n}}{r}-\frac{1}{r}\biggl]\)

\(Zn = Z0 × (1 - r)n -\frac{d(1-r)^{n}}{r}+\frac{d}{r}\)

\(Zn = (1 - r)n \biggl[Z_{0}-\,\frac{d}{r}\biggl]+\frac{d}{r}\)

In order to maintain the convention, the nth term of a sequence is denoted by *T _{n}* and the first term is denoted by

Therefore -

\(Tn = (1 - r)n \biggl[a-\frac{d}{r}\biggl]+\frac{d}{r}\) (equation-3)

To verify the reliability of the generalized formula of a function which involves both the arithmetic and geometric progression, the number of trees (estimated) for the years 2025, will be calculated using equation (3) -

\(Tn = (1 - r)n \biggl[a-\frac{d}{r}\biggl]+\frac{d}{r}\)

\(T5 = (1 - 0. 05)5\biggl[8200-\frac{600}{0.05}\biggl]+\frac{600}{0.052}\)

*T*_{5 }= (0.95)^{5} [8200 - 12000] + 12000

*T*_{5 }= (0.7737809375) × [- 3800] + 12000

*T*_{5 }= - 2940.367 + 12000

*T*_{5 }= 9057.63 ≈ 9060

(Verified)

To find the estimated number of trees after 5 years and hence, determination of a generic formula of number of trees in *n*^{th} year (n^{th} term of a sequence) if the growth rate in the number of trees of the forest is an arithmetic progressive function and the death rate is a geometric progressive function.

The estimated number of trees in the year 2025 will be approximately 9060. Henceforth, the formula of an event which involves both the geometric progression followed by arithmetic progression in every step is found to be - \(T_n = (1 - r)^n \biggl[a-\frac{d}{r}\biggl]+\frac{d}{r},\) where, *d* is the common difference, *r* is the common ration and *T _{n}* is the

- When moved forward from 2020 to 2025, the number of trees has increased from 8200 to 9060.
- An increasing and linear relationship has been obtained between the plant count and the number of years.
- The equation of the obtained trend between the considered dependent variable and the independent variable is:
*y*= 171.91*x*- 339052 where*y*represents the number of plants and*x*represents years. - Due to absence of any significant data point, the obtained relationship can be assumed to be stable. It is also verified by such a high value of the regression correlation coefficient of 0.99.
- Any event which operates on both arithmetic progression as well as geometric progression, simultaneously, varies linearly.

- The estimated number of plants has been calculated in many different ways. Firstly, by manual calculation (step-wise calculation). Secondly, by graphical representation and equation of trend. Thirdly, by developing a generalized equation of a sequence which involves both the geometric progression followed by arithmetic progression in every step.
- The value of regression correlation coefficient of the graph has been obtained to be 0.99. Being a value of correlation coefficient close to 1, it indicates the stability and the strength of correlation, making the exploration more coherent.
- The derived formula of a sequence which involves both the geometric progression followed by arithmetic progression in every step has been verified in section 4.4.

There is no way of verifying the reliability of the obtained data. However, that would not affect the exploration as such; because the purpose of this exploration is to find a generalized formula of any event that follows a geometric sequence followed by an arithmetic operation. Thus, values of any discrepancy in common ratio and common difference would not affect generalized equation derived at the end of the exploration.

To extend the exploration even further, any event that comprises combination of arithmetic, geometric and harmonic sequence could be investigated. Similar to the above exploration, in the future aspect of the exploration, a generalized formula of the n^{th} term of the sequence would be determined which involves arithmetic progression, geometric progression and harmonic progression simultaneously at the same time.

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