Math Problem Statement

Determine the summation notation for the series 3 + 9 + 27+81+243

Solution

The given series is:

3+9+27+81+2433 + 9 + 27 + 81 + 243

This is a geometric series where the first term a=3a = 3 and the common ratio r=3r = 3, because each term is multiplied by 3 to get the next term.

General Formula for a Geometric Series

A geometric series can be written as:

Sn=a+ar+ar2++arn1S_n = a + ar + ar^2 + \dots + ar^{n-1}

Where:

  • aa is the first term,
  • rr is the common ratio,
  • nn is the number of terms.

For the given series:

  • a=3a = 3,
  • r=3r = 3,
  • Number of terms n=5n = 5.

Thus, the summation notation for the given series is:

k=0433k\sum_{k=0}^{4} 3 \cdot 3^k

Or equivalently:

k=043k+1\sum_{k=0}^{4} 3^{k+1}

This represents the sum of the terms 3+9+27+81+2433 + 9 + 27 + 81 + 243.

Would you like any further details or clarifications?

Here are five related questions you could explore:

  1. What is the sum of this geometric series?
  2. How does the formula change for an infinite geometric series?
  3. What happens if the common ratio is less than 1?
  4. Can you generalize this series for any number of terms?
  5. How do you find the sum of a finite geometric series?

Tip: In geometric series, the sum can be found easily using the formula Sn=a(1rn)1rS_n = \frac{a(1 - r^n)}{1 - r} if r1|r| \neq 1.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Geometric Series

Formulas

General formula for geometric series: S_n = a + ar + ar^2 + ... + ar^(n-1)
Summation notation: ∑_{k=0}^{n-1} a * r^k

Theorems

Geometric series summation formula

Suitable Grade Level

Grades 9-12