急急急!!指數率的一些問題...

公式：
1 + 2 + 3 +... + n = n(n+1)/2
1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6
1³ + 2³ + 3³ + ... + n³ = n²(n+1)²/4
等比數列的和 S(n) = a(1 - r^n)/(1 - r)


5² + 10² + 15² + ... + 500²
= 5² x (1² + 2² + 5² + ... + 100²)
= 25 x (100 x 101 x 201)/6
= 8458750


3³ + 6³ + 9³ + ... + 300³
= 3² x (1³ + 2³ + 3³ + ... + 100³)
= 9x (101² x 101²)/4
= 229522500


15³ + 16³ + 17³ + ... + 40³
= (1³ + 2³ + 3³ + ... + 40³)- (1³ + 2³ + 3³ + ... + 14³)
= (40² x 41² / 4) - (14² x 15² /4)
= 661375


1 + (1/3) + (1/3^2) + (1/3^3) + ... + (1/3^6)
= 1 x [1 - (1/3)^7] / [1 - (1/3)]
= (2186/2187) / (2/3)
= 1093/729


(1/2) + (3/4) + (7/8) + (15/16) + ... + 第12項
= [1 - (1/2)] + [1 - (1/2)^2] + [1 - (1/2)^3] + ... + [1 - (1/2)^12]
= 12 - [(1/2) + (1/2)^2 + (1/2)^3 + ... + (1/2)^12]
= 12 - (1/2)[1 - (1/2)^12]/[1 - (1/2)]
= 11又1/4096


(3/2) + (5/4) + (9/8) + ... + 第10項
= [1 + (1/2)] + [1 + (1/2)^2] + [1 + (1/2)^3] + ... + [1 + (1/2)^10]
= 10 + [(1/2) + (1/2)^2 + (1/2)^3 + ... + (1/2)^10]
= 10 + (1/2)[1 - (1/2)^10]/[1 - (1/2)]
= 10又1023/1024


10x11 + 11x12 + 12x13 + ... + 29x30
= (1 + 9)(1 + 10) + (2 + 9)(2 + 10) + (3 + 9)(3 + 10) + ... + (20 + 9)(20 + 10)
= (1² + 19x1 + 90) + (2² + 19x2 + 90) + (3² + 19x3 + 90) + ... + (20² + 19x20 + 90)
= (1² + 2² + 3² + ... + 20²) + 19(1 + 2 + 3 + .. 20) + 90x20
= (20x21x41/6) + (19x20x21/2) + 1800
= 8660


=====
2x4 + 4x6 + 6x8 + ...... + 28x30
= 2x(2 + 2) + 4x(4 + 2) + 6x(6 + 2) + ...... 28x(28 + 2)
= (2² + 4) + (4² + 8) + (6² + 12) + ...... + (28² +56)
= 2²(1²+ 2² + 3² + ...... + 14²) + 4(1 + 2 + 3 + ..... + 14)
=(4 x 14 x 15 x 29 / 6) + (4 x 14 x 15 / 2)
= 4480


=====
10x20 + 20x30 + 30x40 + ...... + 90x100
= 10x(10 + 10) + 20x(20 + 10) + 30x(30 + 10) + ...... + 90x(90 + 10)
= (10² + 100) + (20² + 200) + (30² + 300) + ...... +(90² + 900)
= (10² + 20² + 30² + ...... + 90²) + (100 + 200 + 300 + ...... + 900)
= 10² x (1² + 2² + 3² + ...... + 9²) + 100 x (1 + 2+ 3 + ...... + 9)
= [100 x 9 x (9 + 1) x (2 x 9 + 1) / 6] + [100 x 9 x (9 + 1) / 2]
= 33000

嘛，我還是決定不移除了~~

5^2+10^2+15^2+...+500^2
先找出底數的規律
a1=5
a2=10
a3=15
...
an=5+(n-1)*5=5n
a100=500
利用Σ做計算
5^2+10^2+15^2+...+500^2
=Σ_(n=1~100)(5n)^2
=Σ_(n=1~100)(25n^2)
=25Σ_(n=1~100)(n^2)
=25*100*101*201/6
=8458750
在做Σ的運算時,若不是從第一項開始,則不可套公式!