Non-linear equations and sequences

**juangohan9** · August 4th 2006, 12:49 AM

These assignments seem to be getting harder as they go on.

Hopefully, these'll be the last for a while.

Thanks again...

Solve the system of equations using any method.

logx(5y) = 3
logx(y) = 2

x=? and y=?

A rectangle has a perimeter of 72 centimeters and an area of 320 square centimeters. Find the dimensions of the rectangle.

Find the general, or nth, term of the arithmetic sequence given the first term a1 and the common difference d.
a1 = - 1, d = - 7/10.

Find the sum.
3 + 0 - 3 - ... - 75 = ...

Dr. Wesely contributes 500 from her paycheck (weekly) to a tax-deferred investment account. Assuming the investment earns 6.5% and is compounded weekly, how much will be in the account in 48 weeks?

THANKS.

**galactus** · August 4th 2006, 02:48 AM

#1:

$log_{x}(5y)=3$
$log_{x}(y)=2$

$5y=x^{3}$ ....[1]
$y=x^{2}$ ....[2]

Sub [2] into [1]:

$5(x^{2})=x^{3}$

$x=5$ .

#2:

perimeter: 2x+2y=72

Area: xy=320

Solve the system.

**earboth** · August 4th 2006, 03:28 AM

Originally Posted by juangohan9

...
Find the general, or nth, term of the arithmetic sequence given the first term a1 and the common difference d.
a1 = - 1, d = - 7/10.

Hello, juangohan,

I assume that you know the equation of an arithmetic sequence:
$a_n=a_1+(n-1)\cdot d,\ \ \ \ n=1; 2; 3; ...$ . Plug in the values you know:

$a_n=(-1)+(n-1)\cdot \left(-\frac{7}{10}\right),\ \ \ \ n=1; 2; 3; ...$

Originally Posted by juangohan9

Find the sum.
3 + 0 - 3 - ... - 75 = ...

the sum of an arithmetic serie is:
$s_n=\frac{n\cdot (a_1+a_n)}{2}$ .
First you have to calculate how many summands you actually have (27), then plug in all other given values:
$s_n=\frac{27\cdot (3+(-75))}{2}=-972$

Greetings

EB

**ThePerfectHacker** · August 4th 2006, 07:17 AM

Originally Posted by juangohan9

3 + 0 - 3 - ... - 75 = ...

Here is another way which I always use for arithmetic sums.
$3-3(0),3-3(1),3-3(2),3-3(3),...,3-3(n)$
Find the last term,
$3-3n=75$
Thus,
$-3n=-72$
Thus,
$n=24$
Thus, this is the following summation,
$\sum_{k=1}^{24} 3-3k=3(24)-3\left( \frac{24(25)}{2} \right)$

**Soroban** · August 4th 2006, 10:46 AM

Hello, juangohan9!

Solve the system: . $\begin{array}{cc}\log_x(5y) \,= \,3 \\ \log_x(y) \,= \,2\end{array}$

The first equation is: . $\log_x(5) + \underbrace{\log_x(y)}\;=\;3$

Substitute the second: . $\log_x(5) + 2\;=\;3$

We have: . $\log_x(5)\:=\:1\quad\Rightarrow\quad x^2\:=\:1\quad\Rightarrow\quad\boxed{ x= 5}$

Dr. Wesely contributes $500 from her weekly paycheck to an investment account.
Assuming the investment earns 6.5% and is compounded weekly,
how much will be in the account in 48 weeks?

This is an Annuity problem and requires a formula: . $A \;= \;D\,\frac{(1+i)^n - 1}{i}$

. . where $D$ = periodic deposit, $i$ = periodic interest rate, $n$ = number of periods.

We have: . $D = 500,\;i = \frac{0.065}{52},\;n = 48$

Therefore: . $A \;= \;500\,\frac{\left(1 + \frac{0.065}{52}\right)^{48} - 1}{\frac{0.065}{52}}\;\approx\;\boxed{\$24,718.70}$

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