Questions: Express the sum using summation notation. Use 1 as the lower limit of summation and i for of summation. 5+5^2+5^3+...+5^6 5+5^2+5^3+...+5^6=∑

Express the sum using summation notation. Use 1 as the lower limit of summation and i for of summation. 5+5^2+5^3+...+5^6 5+5^2+5^3+...+5^6=∑
Transcript text: Express the sum using summation notation. Use 1 as the lower limit of summation and i for of summation. \[ \begin{array}{r} 5+5^{2}+5^{3}+\cdots+5^{6} \\ 5+5^{2}+5^{3}+\cdots+5^{6}=\sum \end{array} \]
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Solution

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Solution Steps

To express the given sum using summation notation, we need to identify the pattern in the series. The series given is \(5 + 5^2 + 5^3 + \cdots + 5^6\). This can be written as a summation where the base is 5 and the exponent ranges from 1 to 6.

Solution Approach
  1. Identify the base and the range of exponents in the series.
  2. Use summation notation to express the series.
Step 1: Identify the Series

The series given is \(5 + 5^2 + 5^3 + \cdots + 5^6\). This can be expressed in summation notation as: \[ \sum_{i=1}^{6} 5^i \]

Step 2: Calculate the Summation

To find the value of the summation, we evaluate: \[ \sum_{i=1}^{6} 5^i = 5^1 + 5^2 + 5^3 + 5^4 + 5^5 + 5^6 \] Calculating each term:

  • \(5^1 = 5\)
  • \(5^2 = 25\)
  • \(5^3 = 125\)
  • \(5^4 = 625\)
  • \(5^5 = 3125\)
  • \(5^6 = 15625\)

Adding these values together: \[ 5 + 25 + 125 + 625 + 3125 + 15625 = 19530 \]

Final Answer

The value of the summation is \(\boxed{19530}\).

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