solve the sigma notation where N=1, 6 terms and the geometric sequence formula is 7^3-i

You provided a sigma notation problem involving a geometric sequence. The sum is described as having 6 terms, with \( N = 1 \) as the lower limit, and the formula is \( 7^{3-i} \).

To express this properly:

\[
\sum_{i=1}^{6} 7^{3-i}
\]

This summation involves computing the sum of 6 terms, where each term is \( 7^{3-i} \) for values of \( i \) from 1 to 6.

### Steps to solve:

1. Evaluate each term of the sequence:
   - For \( i = 1 \): \( 7^{3-1} = 7^2 = 49 \)
   - For \( i = 2 \): \( 7^{3-2} = 7^1 = 7 \)
   - For \( i = 3 \): \( 7^{3-3} = 7^0 = 1 \)
   - For \( i = 4 \): \( 7^{3-4} = 7^{-1} = \frac{1}{7} \)
   - For \( i = 5 \): \( 7^{3-5} = 7^{-2} = \frac{1}{49} \)
   - For \( i = 6 \): \( 7^{3-6} = 7^{-3} = \frac{1}{343} \)

2. Sum these values:

\[
49 + 7 + 1 + \frac{1}{7} + \frac{1}{49} + \frac{1}{343}
\]

### Now let's compute the sum numerically:

\[
\sum_{i=1}^{6} 7^{3-i} = 49 + 7 + 1 + 0.14286 + 0.02041 + 0.00292
\]

\[
= 57.16619
\]

Thus, the sum of the series is approximately \( 57.17 \).

### Would you like further details or a breakdown of each step?

### 5 Relative Questions:
1. How is sigma notation applied in arithmetic or geometric sequences?
2. What is the difference between arithmetic and geometric sequences?
3. How do negative exponents affect the behavior of a geometric sequence?
4. Can sigma notation be used for more complex series, like alternating or exponential sums?
5. How would you handle infinite geometric series in sigma notation?

### Tip:
When summing terms with negative exponents, it's helpful to express them as fractions for clearer understanding and accurate calculation.

Explore how to solve a sigma notation problem involving a geometric sequence formula, 7^(3-i), with a step-by-step solution. This problem covers sigma notation applications, geometric sequences, and exponentiation. Suitable for advanced high school students.

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