Summation and geometric sums

So the common ratio is $r= -1/2$.In summary, The sum S can be represented as 3-3/2+3/4-3/8+3/16-3/32+...-3/128. The integers a and n are 3 and 7 respectively, and the rational number k is -1/2. Using the geometric sum formula, the sum can be calculated as 3 times the sum from j=0 to 7 of (-1/2)^j. The common ratio is -1/2.
  • #1
Kola Citron
1
0
Hey!

I'm stuck again and not sure how to solve this question been at it for a few hours. Any help is appreciated as always.

Q: (1) Let the sum S = 3- 3/2 + 3/4 - 3/8 + 3/16 - 3/32 +...- 3/128. Determine integers a , n and a rational number k so that...(Image)

r/askmath - Summation and geometric sums

(2 )And then calculate S using the geometric sum formula.

Thank you!
 
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  • #2
common ratio is $r = -\dfrac{1}{2}$

note $128 = 2^7$

first term is $a = 3$

$\displaystyle S = 3 \sum_{j=0}^7 \left(-\dfrac{1}{2}\right)^j$

you can calculate the sum ...
 
  • #3
Given that it is a "geometric sum", a+ ar+ ar^2+ ...,, you can determine r, the "common ratio" by just dividing the second term by the first: ar/a= r. In this problem that is (-3/2)/3= -1/2.
 

What is summation?

Summation is the process of adding together a sequence of numbers. It is often represented using the Greek letter sigma (∑) and is used to find the total value of a series of terms.

What is a geometric sum?

A geometric sum is a type of summation where each term in the series is multiplied by a constant ratio. This type of sum is also known as a geometric series.

How do you find the sum of a finite geometric series?

To find the sum of a finite geometric series, you can use the formula S = a(1-r^n)/(1-r), where a is the first term, r is the common ratio, and n is the number of terms in the series.

What is the difference between an arithmetic sum and a geometric sum?

An arithmetic sum is a series where each term is added by a constant difference, while a geometric sum is a series where each term is multiplied by a constant ratio. In other words, the pattern in an arithmetic sum is linear, while the pattern in a geometric sum is exponential.

What are some real-life applications of summation and geometric sums?

Summation and geometric sums are used in various fields such as finance, physics, and computer science. For example, in finance, geometric sums are used to calculate compound interest, while in physics, summation is used to find the total distance traveled by an object. In computer science, summation is used in algorithms and programming to find the total number of iterations or operations.

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