M1 Maths chp 0.3 Notation

10. (a)
r(r + 1) - (r - 1)r
= r [(r + 1) - (r - 1)]
= r (r + 1 - r + 1)
= 2r

10. (b)
r = [r(r + 1) - (r - 1)r] / 2

[50Σ(r = 10)] (r)
= [50Σ(r = 10)] {[r(r + 1) - (r - 1)r] / 2}
= (1/2) [10(11) - 9(10) + 11(12) - 10(11) +12(13) - 11(12) + .... + 49(50) - 48(49) + 50(51) - 49(50)]***
= (1/2) [-9(10) + 50(51)]
= 1230

***上面10(11) 與 -10(11) 會互相抵消，
11(12) 亦會與 -11(12) 互相抵消
如此類推
最後會餘下 -9(10) 及 +50(51)

--------------------------------------------------------------------------------------------------------

11.
[nΣ(r = 1)] (r - 1)(r + 1)
= [nΣ(r = 1)] r^2 - 1
= [nΣ(r = 1)] r^2 - [nΣ(r = 1)] 1
= (1/6)n(n + 1)(2n + 1) - (n - 1 + 1)
= (1/6)n(n + 1)(2n + 1) - n


when n = 100

[100Σ(r = 10)] (r - 1)(r + 1)
= [100Σ(r = 1)] (r - 1)(r + 1) - [9Σ(r = 1)] (r - 1)(r + 1)

[100Σ(r = 1)] (r - 1)(r + 1)
= (1/6)(100)(100 + 1)(200 + 1) - 100
= 338250

[9Σ(r = 1)] (r - 1)(r + 1)
= (1/6)(9)(9 + 1)(18 + 1) - 9
= 276

[100Σ(r = 10)] (r - 1)(r + 1)
= [100Σ(r = 1)] (r - 1)(r + 1) - [9Σ(r = 1)] (r - 1)(r + 1)
= 338250 - 276
= 337974