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3 years ago

19 = 15 w- 4(3 w-1) solve for w

Mathematics

1 answer:

Gnom [1K]3 years ago

8 0

Step-by-step explanation:

Simplifying

19 = 15w + -4(3w + -1)

Reorder the terms:

19 = 15w + -4(-1 + 3w)

19 = 15w + (-1 * -4 + 3w * -4)

19 = 15w + (4 + -12w)

Reorder the terms:

19 = 4 + 15w + -12w

Combine like terms: 15w + -12w = 3w

19 = 4 + 3w

Solving

19 = 4 + 3w

Solving for variable 'w'.

Move all terms containing w to the left, all other terms to the right.

Add '-3w' to each side of the equation.

19 + -3w = 4 + 3w + -3w

Combine like terms: 3w + -3w = 0

19 + -3w = 4 + 0

19 + -3w = 4

Add '-19' to each side of the equation.

19 + -19 + -3w = 4 + -19

Combine like terms: 19 + -19 = 0

0 + -3w = 4 + -19

-3w = 4 + -19

Combine like terms: 4 + -19 = -15

-3w = -15

Divide each side by '-3'.

w = 5

Simplifying

w = 5

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A bank account earns 2.5% interest per year. If the account starts with $500, approximately how many years will it take for the

belka [17]

Answer:

Step-by-step explanation:

Since we have an amount in the future of 750, we are going to use Future value formula; FV = PV (1+r)^t

where PV= Initial amount deposited

r= interest rate or discount rate

t = total duration of the investment

FV= 750

PV=500

r = 2.5% or 0.025 as a decimal

t = ?

Next, plug in the numbers into the formula;

750 = 500* (1+0.025)^t

divide both sides by 500;

750/500 = 1.025^t

Introduce <em>ln</em> on both sides

ln 1.5 = ln $1.025^{t}$

ln 1.5 = t ln 1.025

0.4054651 = 0.0246926 t

Divide both sides by 0.0246926 to solve for t;

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t = 16.42

Therefore it will take 16.42 years

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4 years ago

Write the number in standard form 7.5 × 10^9

frutty [35]

7.5 x 10⁹

For standard form, move the decimal 9 places. Since the 9 is positive, move it to the right. (If it was negative 9, you would move it to the left)

Answer: 7,500,000,000

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3 years ago

Determine the above sequence converges or diverges. If the sequence converges determine its limit

marshall27 [118]

Answer:

This series is convergent. The partial sums of this series converge to $\displaystyle \frac{2}{3}$ .

Step-by-step explanation:

The $n$ th partial sum of a series is the sum of its first $n\!\!$ terms. In symbols, if $a_n$ denote the $n\!$ th term of the original series, the $\! n$ th partial sum of this series would be:

$\begin{aligned} S_n &= \sum\limits_{k = 1}^{n} a_k \\ &= a_1 + a_2 + \cdots + a_{k}\end{aligned}$ .

A series is convergent if the limit of its partial sums, $\displaystyle \lim\limits_{n \to \infty} S_{n}$ , exists (should be a finite number.)

In this question, the $n$ th term of this original series is:

$\displaystyle a_{n} = \frac{{(-1)}^{n+1}}{{2}^{n}}$ .

The first thing to notice is the ${(-1)}^{n+1}$ in the expression for the $n$ th term of this series. Because of this expression, signs of consecutive terms of this series would alternate between positive and negative. This series is considered an alternating series.

One useful property of alternating series is that it would be relatively easy to find out if the series is convergent (in other words, whether $\displaystyle \lim\limits_{n \to \infty} S_{n}$ exists.)

If $\lbrace a_n \rbrace$ is an alternating series (signs of consecutive terms alternate,) it would be convergent (that is: the partial sum limit $\displaystyle \lim\limits_{n \to \infty} S_{n}$ exists) as long as $\lim\limits_{n \to \infty} |a_{n}| = 0$ .

For the alternating series in this question, indeed:

$\begin{aligned}\lim\limits_{n \to \infty} |a_n| &= \lim\limits_{n \to \infty} \left|\frac{{(-1)}^{n+1}}{{2}^{n}}\right| = \lim\limits_{n \to \infty} {\left(\frac{1}{2}\right)}^{n} =0\end{aligned}$ .

Therefore, this series is indeed convergent. However, this conclusion doesn't give the exact value of $\displaystyle \lim\limits_{n \to \infty} S_{n}$ . The exact value of that limit needs to be found in other ways.

Notice that $\lbrace a_n \rbrace$ is a geometric series with the first term is $a_0 = (-1)$ while the common ratio is $r = (- 1/ 2)$ . Apply the formula for the sum of geometric series to find an expression for $S_n$ :

$\begin{aligned}S_n &= \frac{a_0 \cdot \left(1 - r^{n}\right)}{1 - r} \\ &= \frac{\displaystyle (-1) \cdot \left(1 - {(-1 / 2)}^{n}\right)}{1 - (-1/2)} \\ &= \frac{-1 + {(-1 / 2)}^{n}}{3/2} = -\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\end{aligned}$ .

Evaluate the limit $\displaystyle \lim\limits_{n \to \infty} S_{n}$ :

$\begin{aligned} \lim\limits_{n \to \infty} S_{n} &= \lim\limits_{n \to \infty} \left(-\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\right) \\ &= -\frac{2}{3} + \frac{2}{3} \cdot \underbrace{\lim\limits_{n \to \infty} \left[{\left(-\frac{1}{2}\right)}^{n} \right] }_{0}= -\frac{2}{3}\end{aligned}}_$ .