It looks like you want to compute the double integral
over the region <em>D</em> with the unit circle <em>x</em> ² + <em>y</em> ² = 1 as its boundary.
Convert to polar coordinates, in which <em>D</em> is given by the set
<em>D</em> = {(<em>r</em>, <em>θ</em>) : 0 ≤ <em>r</em> ≤ 1 and 0 ≤ <em>θ</em> ≤ 2<em>π</em>}
and
<em>x</em> = <em>r</em> cos(<em>θ</em>)
<em>y</em> = <em>r</em> sin(<em>θ</em>)
d<em>x</em> d<em>y</em> = <em>r</em> d<em>r</em> d<em>θ</em>
Then the integral is
There is no common ratio or common difference. The first three terms increases more slowly than a cubic, but the last three increase more quickly than the cube of n. The pattern is not obvious to me. When the sequence is cast in recursive terms, you get
... a[n] = (48/13)a[n-2] + (59/39)a[n-1] . . . . a[1] = 2, a[2] = 9
The next term using this rule is 174 34/39, not an integer.
The coefficients p and q for a[n-2] and a[n-1] can be found from
_____
Any sequence of 4 numbers can be matched by a polynomial of degree 3 or less. Here, a calculator's polynomial regression function tells us the rule could be
... a[n] = 4.5n³ -24.5n² +49n -27
Using this rule, the next two terms are 168 and 357.
Answer:
1 year: $2060
2 years: $2121.80
3 years: $2185.45
Step-by-step explanation:
Compound interest formula is A = P(1 + ) where A is the final amount, P is the initial principal balance, r is the interest rate, n is the number of times interest applied per time period, and t is the number of time periods elapsed. In our case, P would be equal to 2000 dollars, r would be equal to 0.03, for 3 percent, and our n value would just be one, so the final equation is:
First, let's evaluate t for 1, as in one year.
= 2000 x 1.03 = 2060
Two years: 2000 * 1.03 squared = 2121.80
Three years: 2000 * 1.03^3 = 2185.45!
Hope this helps!