F.6 math arithmetric sequence

(a) Let the first term be a and the common difference be d.
The general term is T(n)=a+(n-1)d
The second term is:
a+d=75 . . . . . . (1)
The fifth term is:
a+4d=51 . . . . . (2)

(2)-(1):
3d=-24
d=-8
Hence, a=83

T(n)=83+(n-1)(-8)=91-8n

T(n)>0
91-8n>0
n<91/8~11.38

The largest integral value of n is 11.
T(11)=91-8X11=3

The smallest positive term of the sequence is the 11th term. Its value is 3.


(b) Sn=n[2X83+(n-1)(-8)]/2=n(87-4n)

Sw ≧ Si, where i=1, 2, 3, ... , w, ...
Amount all Si, Sw is the largest

Method 1:
Obviously, the sum of ALL positive terms of the sequence is the largest.
From (a), Sw is the largest when w=11.

S11=11X(87-4X11)=473


Method 2:
Sw=w(87-4w)
=-4(w^2-87w/4)
=-4[w^2-87w/4+(87/8)^2-(87/8)^2]
=-4[(w-87/8)^2-(87/8)^2]

Sw is maximum when w=87/8~10.88 (correct to 4 sig. fig.)
w is a positive integer
w=11

S11=-4[(11-87/8)^2-(87/8)^2]=473