 # Proof of Sum of Geometric Series by Mathematical Induction

Considerations of Sum of Geometric Series The sum of geometric series is defined using r, the common ratio and n, the number of terms. The common could be any real numbers with some exceptions; the common ratio is 1 and 0. If the common ratio is 1, the series becomes the sum of constant numbers, […] # Total Distance Travelled by a Ball Dropped Comes to Rest

A ball is dropped from a height of $16$ metres onto a timber floor and bounces. After each bounce, the maximum height reached by the ball is $80 \%$ of the previous maximum height. Thus, after it first hits the floor, it reaches a height of $12.8$ metres before […] Jane borrows $\100 \ 000$ which is to be repaid in equal monthly instalments. The interest rate is $6 \%$ per annum reducible, compounded monthly. It can be shown that the amount, $\b_n$, owing after the $n$ th repayment is given by the formula: $$b_n = […] # Sum of an Infinite Geometric Series To examine the sum of all the terms of an infinite geometric sequence, we need to consider S_n = \dfrac{u_1(1-r^n)}{1-r} when n gets very large. If \left|r\right|>1, the series is said to be divergent and the sum infinitely large.For instance, when r=2 and u_1=1;S_\infty=1+2+4+8+\cdots is infinitely large. If \left|r\right|<1, or -1 \lt r \lt 1, […] # Geometric Series A \textit{geometric series} is the sum of the terms of a geometric sequence.for example: 1, 2, 4, 8, \cdots , 2048 is a finite geometric sequence. 1+2+4+8+ \cdots +2048 is the corresponding finite geometric series. If we are adding the first n terms of an infinite geometric sequence, we are then calculating a finite geometric […] # Arithmetic Series An \textit{arithmetic series} is the sum of the terms of an arithmetic sequence.For example: 4, 7, 10, 13, \cdots,61 is a finite arithmetic sequence. 4+7+10+13+ \cdots +61 is the corresponding arithmetic series. If the first term is u_{1} and the common difference is d, the terms are:$$u_{1},u_{1}+d,u_{1}+2d,u_{1}+3d,\cdots\( \begin{align} \displaystyle \require{color}u_{1} &= u_{1} \\u_{2} &= u_{1} […] # Sigma Notation Another mathematical device that is widely used in sequences and series is called \textit{sigma notation}. The Greek letter, \sum (capital sigma), is used to indicate the sum of a sequence. For example:\sum_{n=1}^{10}{n^2} = 1^2 + 2^2 + 3^2 + \cdots + 10^2$$The limits of the sum, the numbers on the bottom and top of the […] # Compound Interest of Investment Suppose you invest \2000 in the bank. The money attracts an interest rate of 10\% per annum. The interest is added to the investment each year, so the total interest increases. The percentage increase each year is 10\%, so at the end of the year, you will have 110\% of the value at its start. […] # Geometric Sequence Problems Problems of growth and decay involve repeated multiplications by a constant number, common ratio. We can thus use geometric sequences to model these situations.$$ \begin{align} \displaystyle\require{color} \color{red}u_{n} &= u_{1} \times r^{n-1} \\\require{color} \color{red}u_{n+1} &= u_{1} \times r^{n}\end{align} Example 1 The initial population of chicken on a farm was $40$. The population increased by $5$% […] # Geometric Sequence and Geometric Mean

Geometric Sequence Definition Geometric Sequences are sequences where each term is obtained by multiplying the preceding term by a certain constant factor, which is often called $\textit{common ratio}$. A geometric sequence is also referred to as a $\textit{geometric progression}$. David expects $10$% increase per month to deposit to his account. A $10$% increase per month […]