∴( 1+4d)2=( 1+d)( 1+ 13d),
d = 2(∵d > 0)∴an = 1+(n- 1)×2 = 2n- 1;
b2 = a2 = 3,a5=b3=9,
So the common ratio of geometric series {bn} is q=
b3
b2
=3,
∴bn=b2qn-2=3n- 1
(2)① Proof: √.
c 1
b 1
+
c2
b2
+…+
Communication network (short for Communicating Net)
billion
=an+ 1
When n≥2,
c 1
b 1
+
c2
b2
+…+
cn- 1
bn- 1
= Ann
Subtract two expressions to get.
Communication network (short for Communicating Net)
billion
=an+ 1-an=2(n≥2)。
② cn=2bn=2×3n- 1(n≥2) is obtained from ①.
When n= 1,
c 1
b 1
=a2, ∴c 1=3 does not satisfy the above formula.
∴c 1+c2+…+c20 14=3+2×3 1+2×32+…+2×320 13=3+
6-6×320 13
1-3
=3-3+320 14=320 14