# Thread: Need help for systems and sequences

1. ## Need help for systems and sequences

If Tn=(a+b)n-3,what is the common difference? What is t5?

For the quadratic sequence where a= -2 andb=3, find the general term.Find t10 for the sequence.

For the sequence x-2,2x-1,3x,.... Find the next two terms in the sequence. Find tn.

How many terms are in the sequence -4,-16,-36,-64,.....-1764?

Find two numbers whose sum is m and whose difference is n.

Solve the system for x and y:ax-by=a2+b2
x-y=2b

2. Originally Posted by cinbad
If Tn=(a+b)n-3,what is the common difference? What is t5?
Notice that,
$\displaystyle t_{n+1}=(a+b)(n+1)-3$
$\displaystyle t_n=(a+b)n-3$
Thus,
$\displaystyle t_{n+1}-t_n=(a+b)(n+1)-3-[(a+b)n-3]$
Thus,
$\displaystyle t_{n+1}-t_n=(a+b)(n+1)-(a+b)n$
Thus,
$\displaystyle t_{n+1}-t_n=(a+b)(n+1-n)=(a+b)$ Originally Posted by cinbad
For the sequence x-2,2x-1,3x,.... Find the next two terms in the sequence. Find tn.
Notice, you keep adding $\displaystyle x+1$
Thus,
$\displaystyle t_1=(x-2)+0(x+1)$
$\displaystyle t_2=(x-2)+1(x+1)$
$\displaystyle t_3=(x-2)+2(x+1)$
....
$\displaystyle t_n=(x-2)+(n-1)(x+1)$ Originally Posted by cinbad
How many terms are in the sequence -4,-16,-36,-64,.....-1764?
This, is neither arithmetic nor geometric. I think you made a mistake somewhere. Originally Posted by cinbad
Find two numbers whose sum is m and whose difference is n.
You have,
$\displaystyle \left\{\begin{array}{cc}x+y&=m\\x-y&=n\end{array}\right$
Add the two equations, to get,
$\displaystyle 2x=m+n$
Subtract them to get,
$\displaystyle 2y=m-n$
Thus,
$\displaystyle \left\{\begin{array}{cc}x=&\frac{m+n}{2}\\y=&\frac {m-n}{2}\end{array}\right$

3. For the last question you want to solve, $\displaystyle a\not= b$
$\displaystyle \left\{\begin{array}{cc}ax-by=&a^2+b^2 \mbox{ (1)}\\x-y=&2b \mbox{ (2)}\end{array}\right$
Multiply equation (2) by $\displaystyle a$ then subtract from (1),
$\displaystyle ay-by=a^2-2ab+b^2=(a-b)^2$
Thus,
$\displaystyle y(a-b)=(a-b)^2$
Thus,
$\displaystyle y=a-b$
Similarily,
$\displaystyle x=b-a$

4. Originally Posted by cinbad
...

How many terms are in the sequence -4,-16,-36,-64,.....-1764?
Hello,

the equation of this sequence is:
$\displaystyle a(n)=-(2n)^2$

$\displaystyle 1764=42^2$ therefore you've got 21 summands.

Greetings

EB

5. Originally Posted by cinbad
How many terms are in the sequence -4,-16,-36,-64,.....-1764?
Notice that each term is divisable by 4. Thus the sequence is equivalent to
-4[1, 4, 9, 16, ...441] = -4[1^2, 2^2, 3^2, ..., 21^2]. The second matrix puts the number of terms clearly at 21.

Regards,

Rich B.

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