Find the sum of first 6 terms of the G.P 0.1,0.01,0.001
You asked:
Compute the sum of the first \(6\) terms in the sequence \(0.1, 0.01, 0.001, \ldots\).
MathBot Answer:
$$0.1,0.01,0.001,...$$ The sum of the first 6 terms in this geometric sequence is $\frac{111111}{1000000}$.
The nth term in this sequence is given by the formula:
Explicit Formula: $$a_n=10^{- n}$$
Recursive Formula: $$a_n=\frac{a_{n-1}}{10}, \text{where } a_1=0.1$$
Summation Formula:
Option 1:
$$S_n=\frac{a_1 (r^n - 1)}{r - 1}$$ where $r\neq1$, $r$ is the common ratio, $a_1$ is the 1st term, and $n$ is the is the term number.
Option 2:
$$\begin{aligned} S_n&=\sum_{i=1}^{n} a_{i} \\ &=\sum_{i=1}^{n} 10^{- i} \\ &=\frac{10^{- n} \left(10^{n} - 1\right)}{9} \end{aligned}$$where $n$ is the is the term number.