Write the following series in
sigma notation

Ask by Fowler Parry. in South Africa

Jan 06,2025

Question

Write the following series in sigma notation $$ \frac{1}{3}+\frac{4}{9}+\frac{7}{27}+\frac{10}{81}+\ldots \quad+\frac{22}{6561} $$

UpStudy AI · Accepted Answer

The given series is:

$$
\frac{1}{3} + \frac{4}{9} + \frac{7}{27} + \frac{10}{81} + \ldots + \frac{22}{6561}
$$

Let's analyze the pattern:

1. **Denominators**: The denominators are powers of 3:
   $$
   3^1, 3^2, 3^3, 3^4, \ldots, 3^8 \quad (\text{since } 3^8 = 6561)
   $$
   This suggests the denominator for the $$n$$-th term is $$3^n$$.

2. **Numerators**: The numerators form an arithmetic sequence:
   $$
   1, 4, 7, 10, \ldots, 22
   $$
   This sequence has a first term of 1 and a common difference of 3. The general term for the numerator is:
   $$
   1 + 3(n-1) = 3n - 2
   $$

3. **Number of Terms**: Since the last denominator is $$3^8$$, there are 8 terms in the series.

Combining these observations, the series can be written in sigma notation as:

$$
\sum_{n=1}^{8} \frac{3n - 2}{3^n}
$$

**Final Answer:**

$$
\sum_{n=1}^{8} \frac{\,3\,n\;-\;2\,}{\,3^{\,n}\,}
$$
The series can be written in sigma notation as: $$ \sum_{n=1}^{8} \frac{3n - 2}{3^n} $$

Write the following series in
sigma notation

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