Question -
Answer -
Let P (n) = 1 + 3 + 32 + – – – – + 3n –1 = (3n – 1)/2
Now, For n = 1
P (1) = 1 = (31 – 1)/2 = 2/2 =1
P (n) is true for n = 1
Now, let’s check for P (n) is true for n = k
P (k) = 1 + 3 + 32 + – – – – + 3k – 1 =(3k – 1)/2 … (i)
Now, we have to show P (n) is true for n = k + 1
P (k + 1) = 1 + 3 + 32 + – – – – + 3k =(3k+1 – 1)/2
Then, {1 + 3 + 32 + – – – – + 3k – 1}+ 3k + 1 – 1
= (3k – 1)/2 + 3k using equation (i)
= (3k – 1 + 2×3k)/2
= (3×3 k – 1)/2
= (3k+1 – 1)/2
P (n) is true for n = k + 1
Hence, P (n) is true for all n ∈ N.