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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, […]
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 […]