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Determine whether the geometric series is convergent or divergent. 6 + 5 + 25/6 + 125/36 + ... If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

Answers

Answer 1

Answer:

The $n$ -th term in the series is 6 multiplied by the $(n-1)$ -th power of 5/6:

$a_1=6=6\left(\frac56\right)^(1-1)$

$a_2=5=6\left(\frac56\right)^(2-1)$

$a_3=\frac{25}6=6\left(\frac56\right)^(3-1)$

and so on.

$\displaystyle\sum_(n=1)^\infty6\left(\frac56\right)^(n-1)$

Consider the $N$ -th partial sum,

$S_N=\displaystyle\sum_(n=1)^N6\left(\frac56\right)^(n-1)$

$S_N=6\left(1+\frac56+\cdots+(5^(N-2))/(6^(N-2))+(5^(N-1))/(6^(N-1))\right)$

Multiplying both sides by 5/6 gives

$\frac56S_N=6\left(\frac56+(5^2)/(6^2)+\cdots+(5^(N-1))/(6^(N-1))+(5^N)/(6^N)\right)$

and substracting this from $S_N$ gives

$\frac16S_N=6\left(1-(5^N)/(6^N)\right)$

$S_N=36\left(1-\left(\frac56\right)^N}\right)$

As $N\to\infty$ , it's clear that the sum converges to 36.

Answer 2

Answer:

Final answer:

The geometric series in the question is convergent with a common ratio of 5/6. Using the formula for the sum of an infinite geometric series, the sum of the series is found to be 36.

Explanation:

In mathematics, specifically in series, determining whether a geometric series is convergent or divergent is centered around the common ratio value. In terms of this particular series: 6 + 5 + 25/6 + 125/36 + ..., the common ratio is 5/6. Given this common ratio, it's clear that it falls between -1 and 1. Hence, this geometric series is convergent.

Once we establish it is a convergent series, we can calculate its sum using the formula for the sum of an infinite geometric series: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. Inserting the respective values a = 6 and r = 5/6, we get: S = 6 / (1 - 5/6) = 36. Hence, the sum of this infinite geometric series is 36.

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Holly works at a garage.Mr Bakir brings his car in for a service.
He also buys 2 tyres and a battery.
Holly has this list of costs
• service £110
• tyres £49.99 each
• battery £89
She uses this list to start to work out the bill for Mr Bakir.
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Work out the total bill. (You MUST show your working in the box below)
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Answers

The total bill that Mr Bakir paid was £358.78, using mathematical operations.

How the total bill is determined:

The total bill can be determined using mathematical operations, including multiplication and addition as follows:

The cost of car service = £110

The cost of 2 tyres bought = £99.98 (£49.99 x 2)

The cost of a battery = £89

Sub- total = £298.98

VAT (20%) = £59.80

Total = £358.78

Thus, including 20% VAT, the total bill for Mr. Bakir was £358.78.

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I’ve looked everywhere for the answer, so please help a pimp out

Answers

Answer:

Step-by-step explanation:

What is the solution of the system

Answers

$x - 2y = 4 \n x = 2y + 4 \n 3(2y + 4) + y = 5 \n 6y + 12 + y = 5 \n 7y = - 7 \n y = - 1$
x = 2(-1) + 4
x = 2

Whats y=1/4x-12, solve for x

Answers

Answer:

x=48

Step-by-step explanation:

Determine the sum of the arithmetic series 6 + 11 + 16 +......
91.

Answers

Answer:

873

Step-by-step explanation:

so the equation is: 5x+1

sum is:

$(first \: one \: + \: last \: one)/(2) * quantity \: of \: terms \n$

we have 6( 5×1+1) to 91 (5×18+1)

so we have 18 terms

then:

$(91 + 6)/(2) * 18 = 873$

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers.a) ∀x∃y(x2 = y)

Answers

The statement ∀x∃y(x² = y) is true for all real numbers, as for each real number x, there exists a real number y such that x² = y.

How to determine the truth value of a statement?

The statement ∀x∃y(x² = y) is true as explained below:

For every real number x, there exists a real number y such that x² = y.

For example:

If x = 2, then y = 4 because 2² = 4.

If x = -3, then y = 9 because (-3)² = 9.

Since you can always find a real number y that satisfies the equation x² = y for any real number x, the statement is true for all real numbers in its domain.

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The statement is true. If x and y are both real numbers, the statement is:

"for every x, there exist y such that $x^2=y$ "

This is true, because you can pick any real number, square it, and obtain another real number, y. The relation is not surjective, i.e. we will not use all possible values for y, but it doesn't matter. The statement is only asking to find a value for x^2, which we can always do.

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