How to deduce the area formula of circle

How to deduce the area formula of circle

You can use the circumference of a circle:
Divide the circle into several small sectors and expand into an approximate rectangle. The width of the rectangle is the radius of the circle, and the length of the rectangle is half of the circumference of the circle. Then the area of the rectangle (also the area of the circle) is half of the circumference of the circle multiplied by the radius of the circle
3.14*R*R=3.14*R^2
Given that the sequence {an} satisfies an + 1 + an = 4n-3, when A1 = 2, find SN
a(n+1)+an=4n-3,an+a(n-1)=4*(n-1)-3,
So a (n + 1) - A (n-1) = 4, (n ≥ 2)
a1=2,a2=-1
When n is odd,
an=2+(n-1）/2*4=2n,a(n-1)=-1+(n-1)/2*4=2n-5,
So Sn = (2 + 2n) * (n + 1) / 2 / 2 + (- 1 + 2n-5) * (n-1) / 2 / 2
=n^2-n+2
When n is even,
an=-1+2n,a(n-1)=2+2n,
So Sn = (2 + 2 + 2n) * n / 2 / 2 + (- 1 + 2n-1) * n / 2 / 2 = n ^ 2 + n / 2
From an + 1 + an = 4n-3 (n ∈ n *), an + 2 + an + 1 = 4N + 1 (n ∈ n *)
By subtracting the two formulas, an + 2-An = 4
So the sequence {a 2n - 1} is an arithmetic sequence with the first term of a 1 and the tolerance of 4
The sequence {a 2n} is an arithmetic sequence with the first term a 2 and the tolerance 4
From A2 + A1 = 1, A1 = 2, A2 = - 1
So an = 2n, n = 2k-1
2n-5, n = 2K (K ∈... Expansion)
From an + 1 + an = 4n-3 (n ∈ n *), an + 2 + an + 1 = 4N + 1 (n ∈ n *)
By subtracting the two formulas, an + 2-An = 4
So the sequence {a 2n - 1} is an arithmetic sequence with the first term of a 1 and the tolerance of 4
The sequence {a 2n} is an arithmetic sequence with the first term a 2 and the tolerance 4
From A2 + A1 = 1, A1 = 2, A2 = - 1
So an = 2n, n = 2k-1
2n-5，n=2k（k∈Z）．
① When n is odd, an = 2n, an + 1 = 2n-3. Sn = a1 + A2 + a3 + +an=（a1+a2）+（a3+a4）+… +（an-2+an-1）+an
=1+9+… +（4n-11）+2n=n-12×(1+4n-11)2+2n=2n2-3n+52．
② When n is even, Sn = a1 + A2 + a3 + +an=（a1+a2）+（a3+a4）+… +（an-1+an）═1+9+… +（4n-7）=2n2-3n2．
So Sn = 2n2-3n + 52, n = 2k-1
2n2-3n2, n = 2K (K ∈ z)
time conversion
2 hours 45 minutes = () hours
Urgent, urgent, online, within 5 minutes, first come first served
1 minute = 1 / 60 hour
45 minutes = 3 / 4 hours = 0.75 hours
2 hours 45 minutes = (2.75) hours
2+45/60=2.75
2.75 hours
2.75
2.75 hours
Derivation of area formula of circle
A4 paper full]
(in fact, you don't need to change the A4 paper, but you'd better be more detailed. Thank you
The ancients used the method of cutting circle to calculate the circumference of a circle. Archimedes used regular 96 polygon to get the precision of 3 decimal places; Liu Hui used regular 3072 polygon to get the precision of 5 decimal places; Rudolph used regular 262 polygon to get the precision of 35 decimal places
In the sequence {an}, A1 = 3, an = - A (n-1) - 4N (n ≥ 2, and N ∈ n *), the first n terms and Sn of the sequence {an}
(1) It is proved that the sequence {an + 2n + 1} is an equal ratio sequence, and the general term formula of {an} is obtained
(2) Find SN
Detailed process ~ thank you
（1）
Certificate:
an=-a(n-1)-4n
an+2n+1=-a(n-1)-2n+1=-a(n-1)-2(n-1)-1
(an + 2n + 1) / [a (n-1) + 2 (n-1) + 1] = - 1, is the fixed value
a1+2+1=3+2+1=6
The sequence {an + 2n + 1} is an equal ratio sequence with 6 as the first term and - 1 as the common ratio
an+2n+1=6×(-1)^(n-1)=-6×(-1)^n
an=-2n-1+6×(-1)^(n-1)=-2n-1-6×(-1)^n
(2)
Sn=a1+a2+...+an
=-2(1+2+...n)-n-6×[(-1)^1+(-1)^2+...+(-1)^n]
=-n(n+1)-n-6×(-1)×[(-1)^n-1]/(-1-1)
=-n²-2n-3×[(-1)^n-1]
What is the correct writing format for current unit conversion
Just pay attention to the following units. For example, the common units are a, Ma, μ a
1A=1000mA
1mA=1000μA
Physics friend
How to deduce the formula for calculating the area of a circle
Divide the circle into several approximate triangles and put them together into a rectangle
If the circle is divided into regular polygons with more and more sides, the area of the regular polygon will be closer to the circle, and the ratio of the area of the circle to the square of the radius π will be obtained, and then the area of the circle will be calculated directly
The sequence 〔 an 〕 satisfies an + 1 + an = 4n-3. When A1 = 2, Sn is the sum of the first n terms of the sequence 〔 an 〕, and s 2n + 1 is obtained
A (n + 1) + an = 4N - 3A (n + 1) - 2 (n + 1) + 5 / 2 = - (an - 2n + 5 / 2) let BN = an - 2n + 5 / 2B1 = A1 - 2 + 5 / 2 = 5 / 2B (n + 1) = - BN  BN = 5 / 2 * (- 1) ^ (n + 1) an = BN + 2n - 5 / 2 = 5 / 2 [(- 1) ^ (n + 1) - 1] + 2n when n is odd
Divide parity
How to convert the scale units?
In the international system of units, the standard unit of length is "meter", which is indicated by the symbol "m". 1 mm = 1000 μ M 1 cm = 10 mm 1 decimeter = 10 cm 1 m = 10 decimeter 1 km (km) = 1000 m 1 km = 1000 m 1 m = 10 decimeter 1 decimeter = 10 cm 1 cm = 10 mm 1 micron = 0.000 001 M
How to deduce the area formula of circle
A circle is divided into infinitely many small sectors along its diameter. Because each sector is very small, it can be regarded as a triangle approximately. When these triangles are put together, a rectangle is approximately obtained. The length of the rectangle is half of the circumference of the circle, and the width is equal to the radius of the circle