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4 years ago

Find the sum of a finite geometric sequence from n = 1 to n = 7, using the expression −4(6)n − 1.

Mathematics

1 answer:

Verizon [17]4 years ago

7 0

Answer:

Step-by-step explanation:

The formula of a sum of terms of a gometric sequence:

$S_n=a_1\cdot\dfrac{1-r^n}{1-r}$

a₁ - first term

r - common ratio

We have

$a_n=-4(6)^{n-1}$

Calculate a₁. Put n = 1:

$a_1=-4(6)^{1-1}=-4(6)^0=-4(1)=-4$

Calculate the common ratio:

$r=\dfrac{a_{n+1}}{a_n}\\\\a_{n+1}=-4(6)^{n+1-1}=-4(6)^n\\\\r=\dfrac{-4(6)^n}{-4(6)^{n-1}}=6^n:6^{n-1}\\\\\text{use}\ a^n:a^m=a^{n-m}\\\\r=6^{n-(n-1)}=6^{n-n+1}=6^1=6$

$\text{Substitute}\ a_1=-4,\ n=7,\ r=6:\\\\S_7=-4\cdot\dfrac{1-6^7}{1-6}=-4\cdot\dfrac{1-279936}{-5}=-4\cdot\dfrac{-279935}{-5}=(-4)(55987)\\\\S_7=-223948$

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Answer:

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3 years ago

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The lifetime of a battery in a certain application is normally distributed with mean μ = 16 hours and standard deviation σ = 2 h

DaniilM [7]

Answer:

Probability that a battery will last more than 19 hours is 0.0668.

Step-by-step explanation:

We are given that the lifetime of a battery in a certain application is normally distributed with mean μ = 16 hours and standard deviation σ = 2 hours.

<em>Let X = lifetime of a battery in a certain application</em>

So, X ~ N( $\mu=16,\sigma^{2} =2^{2}$ )

The z-score probability distribution for normal distribution is given by;

Z = $\frac{ X -\mu}{\sigma}$ ~ N(0,1)

where, $\mu$ = mean lifetime = 16 hours

$\sigma$ = standard deviation = 2 hours

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

So, the probability that a battery will last more than 19 hours is given by = P(X > 19 hours)

P(X > 19) = P( $\frac{ X -\mu}{\sigma}$ > $\frac{19-16}{2}$ ) = P(Z > 1.50) = 1 - P(Z $\leq$ 1.50)

= 1 - 0.9332 = 0.0668

<em>Now, in the z table the P(Z </em> $\leq$ <em> x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 1.50 in the z table which has an area of 0.9332.</em>

Hence, the probability that a battery will last more than 19 hours is 0.0668.

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3 years ago

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Dvinal [7]

Answer:

it is detailed solution. please mark brainlest.

Thank you have a nice day