week 09 ap gp exercises possible solutions09周练习可能解决方案.pdfVIP

APGP

Q1AJC2000

Thenthtermofaninfinitegeometricseriesisequaltoonefourthofthesumofallthetermsafter(butnot

including)thenthterm.Showthatthesumtoinfinityofthegeometricseriesisfivetimesitsfirstterm.

Solution:Alternative]

T=1(S−S)

n4∞n

n−11aa[1−rn]

ar=4−

1−r1−r

a

n−1n

4ar=[1−(1−r)]

1−r

4arn−1=S∞[rn]

n−1

4ar4a

S==

∞n

rr−1

a4a

=

1−rr

a

ar=4a(1−r)=5a

1−r

r=4−4r

5r=4S∞=wo)ahns5(

4

r=

5

∴S=4a⋅5=5a(shown)

∞4

Q2VJC2000

The3rdthrd

,11and43termsofanarithmeticprogressionform3consecutivetermsofageometric

progression.Denotingthefirsttermandthecommondifferenceofthearithmeticprogressionbyaandd(d

≠0)respectively,showthata=kd,wherekisaconstanttobedetermined.

Hence,orotherwise,findthecommonratioofthegeometricprogression.

Solution:

T=a+2d

3



T=a+(n−1)dT=a+10d

n11

