W1. Use Mathematical Induction to prove that for all n ≥ 1,

Question

1 +
1
3
+
1
3
2 +
1
33 + · · · +
1
32n =
3
2 �1 − (1
3
)2n+1�

Answer 1

Base case: n = 1
1 + 1/3 = 3/2(1 - 1/9) = 3/2(8/9) = 4/3
This is true.

Inductive Step:
Assume that the statement is true for some k≥1, i.e.,
1 + 1/3 + 1/3^2 + ... + 1/3^2k = 3/2(1 - 1/3^(2k+1))

Now, we will prove that the statement is also true for k+1:
1 + 1/3 + 1/3^2 + ... + 1/3^(2k+2)
= (1 + 1/3 + 1/3^2 + ... + 1/3^(2k)) + 1/3^(2k+1) + 1/3^(2k+2)
= 3/2(1 - 1/3^(2k+1)) + 1/3^(2k+1) + 1/3^(2k+2)
= 3/2(1 - 1/3^(2k+1)) + 1/3^2k * (1/3 + 1/9)
= 3/2(1 - 1/3^(2k+1)) + 1/3^2k * 4/9
= 3/2(1 - 1/3^(2k+1)) + 4/3^(2k+1)
= 3/2(1 - 1/3^(2k+1)) + (3^2 - 1)/3^(2k+1)
= 3/2(1 - 1/3^(2(k+1)+1))

This completes the proof by induction.