設等比數(shù)列的公比為q
∵4a
n-1+a
n+1=4a
n∴4a
n-1+a
n-1q
2=4a
n-1q
∴4+q
2=4q
q=2
a
n=
·2
n-1∴a
1=
, a
2=
·2=
, a
3=
·2
2=
,a
4=
·2
3=
,a
5=
·2
4=
, a
6=
·2
4=
,…
∴sina
1=sin
=
,sina
2=sin
=
,
sina
3=sin
= sin(
)=
,
sina
4=sin(
·2
2)=sin
=sin(
)=-sin
=-
sina
5=sin(
·2
3)=sin
=sin(
)=sin
= sin(
)=
sina
6=sin(
·2
4)=sin
= sin(
)=sin(
)=-sin
=-
一般地,當n≥3時,設n=2k+1(k=1,2,3,…),則
a
n=
·2
n-1=
·2
2k-2=
·4
k-1=
·(1+3)
k-1=
·(1+
3+
3
2+…+
3
k-1)
=
+
(
+
3
1+…+
3
k-2)(規(guī)定
=0)
∴sina
n=sin[
+
(
+
3
1+…+
3
k-2)]=sin
= sin(
)=
,
設n=2k+2(k=1,2,3,…),則
a
n=
·2
n-1=
·2
2k-2=
·4
k-1=
·(1+3)
k-1=
·(1+
3+
3
2+…+
3
k-1)
=
+
(
+
3
1+…+
3
k-2)
∴sina
n=sin[
+
(
+
3
1+…+
3
k-2)]= sin
=sin(
)=-sin
=-
所以,當n≥3時,sina
2k+1+sina
2k+2=0(k=1,2,3,…)
∴sina
1+sina
2+sina
3+…+sina
2014= sina
1+sina
2=
+