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3 years ago

A partial sum of an arithmetic sequence is given. Find the sum.

Mathematics

1 answer:

Nezavi [6.7K]3 years ago

3 0

Answer:

72.3

Step-by-step explanation:

Given that:

The arithmetic sequence:

$\sum \limits ^{14}_{k=0} (3 + 0.26k)$

The first term of the sequence $a_0$ = 3 since k = 0

i.e (3 + 0.26(0)) = 3

The last term of the sequence $a_{14}$ = 6.64

i.e (3+ 0.26(14)

= (3 + 3.64)

= 6.64

Total no of terms = 15 i.e from 0 to 14

∴

The partial sum of the arithmetic sequence = $\dfrac{Total \ no \ of \ terms }{2} \times (a_o+a_{15})}$

$=\dfrac{15 }{2} \times (3+6.64)}$

= 72.3

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A 13-inch piece of ribbon is 33.02 centimeters long. How long will a 24-inch piece be?

m_a_m_a [10]

Answer:

60.96

Step-by-step explanation:

1 inch = 2.54

so to know what 24 inch = you can multiply 24 time 2.54 and you should get the answer

2.54 time 24= 60.96

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3 years ago

Find the distance between P1(4,16degrees) and P2(-2,177degrees) on the polar plane.

bazaltina [42]

Polar coordinates give the distance from the origin and the angle from the positive x axis. Cartesian coordinates give the distance from the x and y axes.

You can draw a right triangle with these values. (see attached)
If you know the r value and theta of that triangle below, you can use trig to find x and y.

Let's convert (4, 16°) to Cartesian coordinates.

Note that since our angle is acute, (in Quadrant I) our sine and cosine will both be positive, as you should be able to derive from the unit circle, where cosine is represented as an x value and sine is represented as a y value.

cosine = adjacent / hypotenuse
cosθ = x/r
cos(16°) = x/4
4cos(16°) = x ≈ 3.84504678375

sine = oppsite / hypotenuse
sinθ = y/r
sin(16°) = y/4
4sin(16°) = y ≈ 1.10254942327<span>

So (4, 16°) </span>⇒ (3.84504678375, 1.10254942327).

Let's convert (-2, 177°) to Cartesian coordinates.
Whenever you have a negative radius, that means to put the point opposite where it would have been if it had a positive radius. (see attached)

In that case, we can essentially add 180° to our current 177° to the same effect. That means that (-2, 177°) = (2, 357°).

Note that since our angle is in Quadrant IV, our cosine will be positive, but our sine will be negative. (as derived from the unit circle) We don't have to worry about this since our calculator figures this for us, but you should pay attention to it if you are converting from Cartesian to polar.

cosine = adjacent / hypotenuse
cosθ = x/r
cos(357°) = x/2
2cos(357°) = x ≈ 1.99725906951

sine = opposite / hypotenuse
sinθ = y/r
sin(357°) = y/2
2sin(357°) = y ≈ -0.10467191248

So (-2, 177°) ⇒ (1.99725906951, -0.10467191248).

Now we must use the distance formula with our two points.
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
$d\approx\sqrt{(1.99725906951-3.84504678375)^2+(-0.10467191248-1.10254942327)^2}$
$d\approx\sqrt{-1.84778771^2+-1.20722134^2}$
$d\approx\sqrt{3.41431942+1.45738336}$
$d\approx\sqrt{4.87170278}$
$\boxed{d\approx2.20719342}$

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3 years ago

3y''-6y'+6y=e*x sexcx

Simora [160]

From the homogeneous part of the ODE, we can get two fundamental solutions. The characteristic equation is

$3r^2-6r+6=0\iff r^2-2r+2=0$

which has roots at $r=1\pm i$ . This admits the two fundamental solutions

$y_1=e^x\cos x$
$y_2=e^x\sin x$

The particular solution is easiest to obtain via variation of parameters. We're looking for a solution of the form

$y_p=u_1y_1+u_2y_2$

where

$u_1=-\displaystyle\frac13\int\frac{y_2e^x\sec x}{W(y_1,y_2)}\,\mathrm dx$
$u_2=\displaystyle\frac13\int\frac{y_1e^x\sec x}{W(y_1,y_2)}\,\mathrm dx$

and $W(y_1,y_2)$ is the Wronskian of the fundamental solutions. We have

$W(e^x\cos x,e^x\sin x)=\begin{vmatrix}e^x\cos x&e^x\sin x\\e^x(\cos x-\sin x)&e^x(\cos x+\sin x)\end{vmatrix}=e^{2x}$

and so

$u_1=-\displaystyle\frac13\int\frac{e^{2x}\sin x\sec x}{e^{2x}}\,\mathrm dx=-\int\tan x\,\mathrm dx$
$u_1=\dfrac13\ln|\cos x|$

$u_2=\displaystyle\frac13\int\frac{e^{2x}\cos x\sec x}{e^{2x}}\,\mathrm dx=\int\mathrm dx$
$u_2=\dfrac13x$

Therefore the particular solution is

$y_p=\dfrac13e^x\cos x\ln|\cos x|+\dfrac13xe^x\sin x$

so that the general solution to the ODE is

$y=C_1e^x\cos x+C_2e^x\sin x+\dfrac13e^x\cos x\ln|\cos x|+\dfrac13xe^x\sin x$

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3 years ago

How much is a 15% tip if the meal was $32?

Digiron [165]

0.15(32) = 4.80 <=====

8 0

3 years ago

Read 2 more answers

If x is a number which expression represents the statement 4 less than the product of 2 and a number

zhannawk [14.2K]

I think the expression that you need is, 4-(2x). Hope it helps!

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3 years ago