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3 years ago

6

Suppose y = 48 + 3(2n - 1) is an explicit representation of an arithmetic sequence for integer values n ≥ 1. Find the xth partia

l sum of the series, as a quadratic function, where x represents the term number.

2 answers:

PSYCHO15rus [73]3 years ago

6 0

Answer: 3x2 + 51x

Step-by-step explanation:

omeli [17]3 years ago

5 0

$a_n=48+3(2n-1)$

The formula of the sum of the arithmetic sequence:
$S_n=\dfrac{a_1+a__n}{2}\cdot n$
calculate:
$a_1=48+3(2\cdot1-1)=48+3=51$
substitute
$S_n=\dfrac{51+48+3(2n-1)}{2}\cdot n=\dfrac{99+6n-3}{2}\cdot n=\dfrac{96+6n}{2}\cdot n=3n^2+48n$
Your answer is:
$\boxed{f(x)=3x^2+48x}$

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Find the point (,) on the curve =8 that is closest to the point (3,0). [To do this, first find the distance function between (,)

ELEN [110]

Question:

Find the point (,) on the curve $y = \sqrt x$ that is closest to the point (3,0).

[To do this, first find the distance function between (,) and (3,0) and minimize it.]

Answer:

$(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})$

Step-by-step explanation:

$y = \sqrt x$ can be represented as: $(x,y)$

Substitute $\sqrt x$ for $y$

$(x,y) = (x,\sqrt x)$

So, next:

Calculate the distance between $(x,\sqrt x)$ and $(3,0)$

Distance is calculated as:

$d = \sqrt{(x_1-x_2)^2 + (y_1 - y_2)^2}$

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$d = \sqrt{x^2 - 6x +9 + x}$

Rewrite as:

$d = \sqrt{x^2 + x- 6x +9 }$

$d = \sqrt{x^2 - 5x +9 }$

Differentiate using chain rule:

Let

$u = x^2 - 5x +9$

$\frac{du}{dx} = 2x - 5$

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$d = \sqrt u$

$d = u^\frac{1}{2}$

$\frac{dd}{du} = \frac{1}{2}u^{-\frac{1}{2}}$

Chain Rule:

$d' = \frac{du}{dx} * \frac{dd}{du}$

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$d' = \frac{2x - 5}{2\sqrt u}$

Substitute: $u = x^2 - 5x +9$

$d' = \frac{2x - 5}{2\sqrt{x^2 - 5x + 9}}$

Next, is to minimize (by equating d' to 0)

$\frac{2x - 5}{2\sqrt{x^2 - 5x + 9}} = 0$

Cross Multiply

$2x - 5 = 0$

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Substitute $x = \frac{5}{2}$ in $y = \sqrt x$

$y = \sqrt{\frac{5}{2}}$

Split

$y = \frac{\sqrt 5}{\sqrt 2}$

Rationalize

$y = \frac{\sqrt 5}{\sqrt 2} * \frac{\sqrt 2}{\sqrt 2}$

$y = \frac{\sqrt {10}}{\sqrt 4}$

$y = \frac{\sqrt {10}}{2}$

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$(x,y) = (\frac{5}{2},\frac{\sqrt{10}}{2}})$

3 0

3 years ago

Katherine invested $86,000 in an account paying an interest rate of 52%

earnstyle [38]

Answer: $4,331‬

Step-by-step explanation:

I will assume that the question meant 5.2% not 52% as this is quite excessive.

Katherine after 13 years.

Convert the variables;

n = 13 years = 13 * 365 = ‭4,745‬ days

r = 5.2% = 5.2/365 = 5.2/365%

Future value = P * ( 1 + r)^n

= 86,000 * (1 + 5.2/365%)⁴⁷⁴⁵

= $169,068

Michael after 13 years;

If compounded continuously, the formula is;

Future value = Pe^rt

= 86,000 * e⁰.⁰⁵ ˣ ¹³

= $164,737

Difference = 169,068 - 164,737

= $4,331‬

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