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noname [10]
3 years ago
14

Write the recursive rule and the iterative rule for the following sequence: 10, 6, 2, -2, -6

Mathematics
1 answer:
barxatty [35]3 years ago
7 0
The recursive rule is
a_n=a_{n-1}+(-4). 
The iterative rule is
a_n=14-4n

The recursive rule is given by the formula 
a_n=a_{n-1}+d, where d is the common difference.  Our common difference is -4, which gives us the recursive rule above.

The iterative rule begins with the formula
a_n=a_1+d(n-1), where a₁ is the first term and d is the common difference.  Our first term is 10 and our common difference is -4:
a_n=10+(-4)(n-1)
\\\text{Using the Distributive Property,}
\\
\\a_n=10+(-4*n)+(-4*-1)
\\
\\a_n=10-4n+4
\\
\\\text{Combining like terms,}
\\
\\a_n=14-4n
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Differentiate <em>V</em> with respect to time <em>t</em> :

d<em>V</em>/d<em>t</em> = 4<em>π</em> <em>r</em> ² d<em>r</em>/d<em>t</em>

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Differentiate <em>A</em> with respect to <em>t</em> :

d<em>A</em>/d<em>t</em> = 8<em>π</em> <em>r</em> d<em>r</em>/d<em>t</em>

<em />

You're given that the volume is increasing at a constant rate, d<em>V</em>/d<em>t</em> = 10 cm³/s, so

10 cm³/s = 4<em>π</em> <em>r</em> ² d<em>r</em>/d<em>t</em>

Solve for d<em>r</em>/d<em>t</em> (the rate of change of the sphere's radius):

d<em>r</em>/d<em>t</em> = 5/(2<em>π</em> <em>r</em> ²) cm³/s

Substitute this into the equation for d<em>A</em>/d<em>t</em> :

d<em>A</em>/d<em>t</em> = 8<em>π</em> <em>r</em> (5/(2<em>π</em> <em>r</em> ²) cm³/s)

d<em>A</em>/d<em>t</em> = 20/<em>r</em> cm³/s

At the moment the radius is <em>r</em> = 5 cm, the surface area is increasing at a rate of

d<em>A</em>/d<em>t</em> = 20/(5 cm) cm³/s = 4 cm²/s

When the surface area is increasing at a rate of d<em>A</em>/d<em>t</em> = 2 cm²/s, the radius <em>r</em> is such that

2 cm²/s = 20/<em>r</em> cm³/s

which happens when the radius is

<em>r</em> = 20/2 (cm³/s) / (cm²/s) = 10 cm

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