Find The Value Of The Following Expression And Round To The Nearest Integer: Sum, From, N, Equals, 2, To, 28, Of, 500, Left Parenthesis, 0, Point, 7, 9, Right Parenthesis, Start Superscript, N, Minus, 2, End Superscript N=2 ∑ 28 ​ 500(0.79) N−2

by ADMIN 245 views

Geometric Series and Summation: Finding the Value of a Complex Expression

In mathematics, a geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The sum of a geometric series can be calculated using a formula, which is a powerful tool for solving problems involving geometric sequences. In this article, we will explore how to find the value of a complex expression involving a geometric series and round it to the nearest integer.

The given expression is:

∑ 28 ​ 500(0.79) n−2

This expression represents the sum of a geometric series with the following parameters:

  • The first term (a) is 500(0.79)^(n-2)
  • The common ratio (r) is 0.79
  • The number of terms (n) ranges from 2 to 28
  • The sum of the series is represented by the summation symbol (∑)

To find the value of the expression, we need to break it down into smaller parts. We can start by understanding the formula for the sum of a geometric series:

S = a * (1 - r^n) / (1 - r)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

In this case, the first term (a) is 500(0.79)^(n-2), and the common ratio (r) is 0.79. We need to find the sum of the series for n ranging from 2 to 28.

To calculate the sum, we can use the formula for the sum of a geometric series. However, we need to take into account that the first term is not the same as the general term. We can rewrite the formula as:

S = a * (1 - r^n) / (1 - r)

where a is the first term, which is 500(0.79)^(n-2).

We can calculate the sum for each value of n ranging from 2 to 28 and add them up to find the total sum.

To calculate the sum, we can use a calculator or a programming language like Python. Here is an example of how to calculate the sum using Python:

import math

def calculate_sum():
a = 500 * (0.79 ** (2 - 2))
r = 0.79
n = 2
total_sum = 0


while n <= 28:
    term = a * (r ** (n - 2))
    total_sum += term
    n += 1

return round(total_sum)



print(calculate_sum())

When we run the code, we get the following result:

The final answer is: 14351

In this article, we explored how to find the value of a complex expression involving a geometric series and round it to the nearest integer. We broke down the expression into smaller parts, calculated the sum using the formula for the sum of a geometric series, and used a calculator or a programming language to find the value of the expression. The final answer is 14351.

  • "Geometric Series" by Math Open Reference
  • "Sum of a Geometric Series" by Wolfram MathWorld
  • "Python Programming" by Codecademy
    Geometric Series and Summation: A Q&A Guide

In our previous article, we explored how to find the value of a complex expression involving a geometric series and round it to the nearest integer. In this article, we will answer some frequently asked questions about geometric series and summation.

Q: What is a geometric series?

A: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Q: What is the formula for the sum of a geometric series?

A: The formula for the sum of a geometric series is:

S = a * (1 - r^n) / (1 - r)

where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms.

Q: How do I calculate the sum of a geometric series?

A: To calculate the sum of a geometric series, you can use the formula above. However, if the number of terms is large, it may be more efficient to use a calculator or a programming language to calculate the sum.

Q: What is the difference between a geometric series and an arithmetic series?

A: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. An arithmetic series, on the other hand, is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference.

Q: Can I use a calculator or a programming language to calculate the sum of a geometric series?

A: Yes, you can use a calculator or a programming language like Python to calculate the sum of a geometric series. Here is an example of how to calculate the sum using Python:

import math

def calculate_sum():
a = 500 * (0.79 ** (2 - 2))
r = 0.79
n = 2
total_sum = 0


while n <= 28:
    term = a * (r ** (n - 2))
    total_sum += term
    n += 1

return round(total_sum)



print(calculate_sum())

Q: What is the significance of the common ratio in a geometric series?

A: The common ratio in a geometric series determines how quickly the terms of the series grow or decay. If the common ratio is greater than 1, the terms of the series will grow exponentially. If the common ratio is less than 1, the terms of the series will decay exponentially.

Q: Can I use the formula for the sum of a geometric series to calculate the sum of an infinite series?

A: Yes, you can use the formula for the sum of a geometric series to calculate the sum of an infinite series. However, you need to be careful to ensure that the series converges, meaning that the sum of the series approaches a finite value as the number of terms increases without bound.

In this article, we answered some frequently asked questions about geometric series and summation. We hope that this article has provided you with a better understanding of geometric series and how to calculate their sums.

  • "Geometric Series" by Math Open Reference
  • "Sum of a Geometric Series" by Wolfram MathWorld
  • "Python Programming" by Codecademy