2^3n – 1 is divisible by 7, for all natural numbers n.
Prove the statements by the Principle of Mathematical Induction :
23n – 1 is divisible by 7, for all natural numbers n.
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Let P(n): 23n – 1 is divisible by 7
Now, P( 1): 23 — 1 = 7, which is divisible by 7.
Hence, P(1) is true.
Let us assume that P(n) is true for some natural number n = k.
P(k): 23k – 1 is divisible by 7.
or, 23k -1 = 7m, m∈ N .......(i)
Now, we have to prove that P(k + 1) is true
P(k+ 1): 23(k+1) – 1
= 23k .23 – 1
= 8(7 m + 1) – 1 [from (i)]
= 56 m + 7
= 7(8m + 1), which is divisible by 7.
Thus, P(k + 1) is true whenever P(k) is true.
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