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Suppose \( n \) consecutive geometric terms are inserted between \( 1 \) and \( 2 \). Write the sum of these \( n \) terms in terms of \( n \). $$ \Large \underbrace{1, \overbrace{u_1, u_2, u_3, \cdots, u_{n}}^{S_n}, 2}_{S_{n+2}} $$ \( \require{AMSsymbols} \displaystyle \begin{align} S_{n+2} &= 1+u_1 + u_2 + u_3 + \cdots […]
Considerations of the 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 […]
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: $$ b_n = […]
To examine the sum of all the terms of an infinite geometric sequence, we need to consider $S_n = \dfrac{u_1(1-r^n)}{1-r}$ when $n$ gets very large. If $\left|r\right|>1$, the series is said to be divergent and the sum infinitely large.For instance, when $r=2$ and $u_1=1$;$S_\infty=1+2+4+8+\cdots$ is infinitely large. If $\left|r\right|<1$, or $-1 \lt r \lt 1$, […]
A $\textit{geometric series}$ is the sum of the terms of a geometric sequence.for example: $1, 2, 4, 8, \cdots , 2048$ is a finite geometric sequence. $1+2+4+8+ \cdots +2048$ is the corresponding finite geometric series. If we add the first $n$ terms of an infinite geometric sequence, we calculate a finite geometric series called the […]
Another mathematical device widely used in sequences and series is $\textit{sigma notation}$. The Greek letter $\sum$ (capital sigma), indicates the sum of a sequence. For example:$$ \large \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 $\sum$, indicate the terms […]
Time payments are calculated based on Geometric Series for reducible compound interests. The geometric series formula is used for handling this situation. Worked on Examples of Geometric Series for Time Payments 1 John took out a loan of \($100 000\) on \(1\)st January \( 2001 \) for home loan. Interest is charged at \(12 \% […]