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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: $$ b_n = […]
To examine the sum of all the terms of an infinite geometric sequence, we need to consider $S_n = \dfrac{u_1(1-r^n)}{1-r}$ when $n$ gets very large. If $\left|r\right|>1$, the series is said to be divergent and the sum infinitely large.For instance, when $r=2$ and $u_1=1$;$S_\infty=1+2+4+8+\cdots$ is infinitely large. If $\left|r\right|<1$, or $-1 \lt r \lt 1$, […]
A $\textit{geometric series}$ is the sum of the terms of a geometric sequence.for example: $1, 2, 4, 8, \cdots , 2048$ is a finite geometric sequence. $1+2+4+8+ \cdots +2048$ is the corresponding finite geometric series. If we are adding the first $n$ terms of an infinite geometric sequence, we are then calculating a finite geometric […]
Another mathematical device that is widely used in sequences and series is called $\textit{sigma notation}$. The Greek letter, $\sum$ (capital sigma), is used to indicate the sum of a sequence. For example:$$\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 […]
Time payments are calculated based on Geometric Series for reducible compound interests. Basically geometric series formula is used for handling this situation. Worked Examples of Geometric Series for Time Payments 1 John takes out a loan of \($100 000\) on 1st January 2001 for a home loan. Interest is charged at 12 % per annum […]