All categories

2 years ago

The arithmetic sequence a_ia

Mathematics

1 answer:

topjm [15]2 years ago

3 0

Answer:

492,800

Step-by-step explanation:

Given ith term of an arithmetic sequence as shown:

ai = a(i-1)+2

and a1 = 5

When i = 2

a2 = a(2-1)+2

a2 = a1+2

a2 = 5+2

a2 = 7

When i = 3

a3 = a(3-1)+2

a3 = a2+2

a3 = 7+2

a3 = 9

It can be seen that a1, a2 and a3 forms an arithmetic progression

5,7,9...

Given first term a1 = 5

Common difference d = 7-5= 9-7 = 2

To calculate the sum of the first 700 of the sequence, we will use the formula for finding the sum of an arithmetic sequence.

Sn = n/2{2a1+(n-1)d}

Given n = 700

S700 = 700/2{2(5)+(700-1)2}

S700 = 350{10+699(2)}

S700 = 350{10+1398}

S700 = 350×1408

S700 = 492,800

Therefore, the sum of the first 700 terms in the sequence is 492,800

You might be interested in

Breanna thought of a number, added 6 to it, multiplied it by 6, subtracted 6 from her answer, and then divided it by 6. She got

ss7ja [257]

Answer:5

Step-by-step explanation:

If you look at the question in reverse you can backtrack your way through. Take 10 and instead of dividing, multiply it by 10, which is 60, add 6 instead of subtracting, which is 66, divide by 6 instead of multiplying, which is 11 and finally subtract 6 instead of adding, which is 5. Use five and try the question over to make sure you calculated correct.

8 0

2 years ago

I'm having trouble with #2. I've got it down to the part where it would be the integral of 5cos^3(pheta)/sin(pheta). I'm not sur

Butoxors [25]

$\displaystyle\int\frac{\sqrt{25-x^2}}x\,\mathrm dx$

Setting $x=5\sin\theta$ , you have $\mathrm dx=5\cos\theta\,\mathrm d\theta$ . Then the integral becomes

$\displaystyle\int\frac{\sqrt{25-(5\sin\theta)^2}}{5\sin\theta}5\cos\theta\,\mathrm d\theta$
$\displaystyle\int\sqrt{25-25\sin^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\sqrt{1-\sin^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\sqrt{\cos^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$

Now, $\sqrt{x^2}=|x|$ in general. But since we want our substitution $x=5\sin\theta$ to be invertible, we are tacitly assuming that we're working over a restricted domain. In particular, this means $\theta=\sin^{-1}\dfrac x5$ , which implies that $\left|\dfrac x5\right|\le1$ , or equivalently that $|\theta|\le\dfrac\pi2$ . Over this domain, $\cos\theta\ge0$ , so $\sqrt{\cos^2\theta}=|\cos\theta|=\cos\theta$ .

Long story short, this allows us to go from

$\displaystyle5\int\sqrt{\cos^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$

to

$\displaystyle5\int\cos\theta\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\dfrac{\cos^2\theta}{\sin\theta}\,\mathrm d\theta$

Computing the remaining integral isn't difficult. Expand the numerator with the Pythagorean identity to get

$\dfrac{\cos^2\theta}{\sin\theta}=\dfrac{1-\sin^2\theta}{\sin\theta}=\csc\theta-\sin\theta$

Then integrate term-by-term to get

$\displaystyle5\left(\int\csc\theta\,\mathrm d\theta-\int\sin\theta\,\mathrm d\theta\right)$
$=-5\ln|\csc\theta+\cot\theta|+\cos\theta+C$

Now undo the substitution to get the antiderivative back in terms of $x$ .

$=-5\ln\left|\csc\left(\sin^{-1}\dfrac x5\right)+\cot\left(\sin^{-1}\dfrac x5\right)\right|+\cos\left(\sin^{-1}\dfrac x5\right)+C$

and using basic trigonometric properties (e.g. Pythagorean theorem) this reduces to

$=-5\ln\left|\dfrac{5+\sqrt{25-x^2}}x\right|+\sqrt{25-x^2}+C$

4 0

2 years ago

Read 2 more answers

Write the rule for finding the coordinates of a point’s 90⁰ clockwise rotation about the origin.

olga nikolaevna [1]

(x,y) -> (y,-x)

hope that help

4 0

2 years ago

I WILL MARK YOU AS BRANLIEST JUST DO THIS PLS ASAP

tensa zangetsu [6.8K]

Answer:

(6t) + (3t) + (4t)

Simplified it would be 13t

3 0

2 years ago

Jamie has a bag containing purple, pink, yellow, and blue marbles. There are a total of 40 marbles in the bag. Jamie randomly se

Ad libitum [116K]

Answer:

1 pink marble

Step-by-step explanation:

If Jamie selected 10 times the pink marble in a total of 420 tries, we can assume that the proportion of pink marble in the total number of marbles is:

10 / 420 = 1/42

So to predict a good number of pink marbles in the total 40 marbles, we just need to multiply the 40 marbles by the proportion found above:

pink marble = 40 * (1/42) = 0.9524

As this number needs to be a whole number, a good assumption is that there is 1 pink marble in the 40 marbles.

8 0

2 years ago