A) n^2 -16n+69 = (n-8)^2 - 64 + 69 = (n-8)^2+5
B) vertex (n,g(n)) = (8, 5). It's a minimum because for any other value of n, (n-8)^2 is positive and it adds (positive sign before the ()^2)
C) n=8 is the axis of symmetry (it is the horizontal value of the vertex)
A. plane. I hope this helps you out! I'm not very good at math myself though haha
Answer:
- a1 = 8; an= an-1 - 2
- an = -3 + 4(n-1)
Step-by-step explanation:
1. The first step in problem solving is to look at the given information. Here, you can see that each number in the sequence is <em>2 less than the one before it</em>.
In the expression ...
an = an-1 - 2
the term an-1 means the previous term, the term just before term an. So, this equation means the term of the sequence is <em>2 less than the one before it</em>. Since this matches the description of the sequence, this recursive relation is part of the correct answer. (The other part is the definition of the first term of the sequence: a1 = 8.)
The recursive definition is ...
__
2. The given sequence has first term a1 = -3, and common difference d = 4. That is, each term is 4 more than the previous one. The explicit formula for an arithmetic sequence is ...
an = a1 +d(n -1)
Filling in the given values gives you the explicit formula ...
3a + 2b = 16....multiply by 2
2a + 3b = 14...multiply by -3
------------------
6a + 4b = 32 (result of multiplying by 2)
-6a - 9b = - 42 (result o multiplying by -3)
-----------------add
-5b = - 10
b = -10/-5
b = 2.....items on shelf B cost $ 2
3a + 2b = 16
3a + 2(2) = 16
3a + 4 = 16
3a = 16 - 4
3a = 12
a = 12/3
a = 4....items on shelf A cost $ 4
an item on shelf A costs (4 - 2) = 2 dollars more then an item on shelf B
Given differential equation, (D<span>4 </span>- 5D3<span> + 5D</span>2<span> + 5D - 6)y = 0</span>
=> For general solution of equation,
Solve D<span>4 </span>- 5D3<span> + 5D</span>2<span> + 5D - 6 = 0</span>
=> D<span>4 </span>- 5D3<span> + 6D</span>2<span> - D</span><span>2 </span>+ 5D - 6 = 0
=> D2<span> (D</span>2<span> - 5D + 6) - (D</span>2<span> - 5D + 6) = 0</span>
=> (D2<span> - 5D + 6)(D</span>2<span> - 1) = 0 ................................(1)</span>
<span>Now </span>
D<span>2 </span><span>- 1 = (D - 1)(D + 1) and </span>
Factors of D2<span> - 5D + 6</span>
D2<span> - 5D + 6 = D</span>2<span> - 2D - 3D + 6</span>
= D(D - 2) - 3(D - 2)
= (D - 3)(D - 2)
Therefore, equation (1) implies
(D2<span> - 5D + 6)(D</span>2<span> - 1) = (D - 3)(D - 2)(D - 1)(D + 1) = 0</span>
=> D = 3, 2, 1, -1 or D = -1, 1,, 2, 3
=> General solution of differential equation is,<span>
=><span> y = C1 e-x + C2 ex + C3 e2x + C4 e3x</span> .
</span>
