Answer:
Step-by-step explanation:
The recursive rule tells you the initial term of the sequence is a1 = -3, and the common difference is d=7. (7 is the value added to one term to get the next term.)
Putting these values into the formula for the explicit rule gives ...
an = a1 +d(n -1)
an = -3 + 7(n -1)
9514 1404 393
Answer:
±√58 ≈ ±7.616
Step-by-step explanation:
The linear combination of sine and cosine functions will have an amplitude that is the root of the sum of the squares of the individual amplitudes.
|x| = √(3² +7²) = √58
The motion is bounded between positions ±√58.
_____
Here's a way to get to the relation used above.
The sine of the sum of angles is given by ...
sin(θ+c) = sin(θ)cos(c) +cos(θ)sin(c)
If this is multiplied by some amplitude A, then we have ...
A·sin(θ+c) = A·sin(θ)cos(c) +A·cos(θ)sin(c)
Comparing this to the given expression, we find ...
A·cos(c) = 3 and A·sin(c) = -7
We know that sin²+cos² = 1, so the sum of the squares of these values is ...
(A·cos(c))² +(A·sin(c))² = A²(cos(c)² +sin(c)²) = A²(1) = A²
That is, A² = (3)² +(-7)² = 9+49 = 58. This tells us the position function can be written as ...
x = A·sin(θ +c) . . . . for some angle c
x = (√58)sin(θ +c)
This has the bounds ±√58.
