Given; a1 = 3 and ak = 7ak–1
⇒ a2 = 7 × a1 = 7 × 3 = 21 = 3.72-1
⇒ a3 = 7 × a2 = 72 × a1 = 49 × 3 = 147 = 3.73-1
⇒ a4 = 7 × a3 = 72 × a2 = 73 × a1 = 343 × 3 = 1029 = 3.74-1
⇒ an = 7an-1 = 3.7n-1
Let am = 7am-1 = 3.7m-1 be true.
⇒ am+1 = 7am+1-1 = 7am = 7.3.7m-1 = 3.7(m+1)-1
⇒ am+1 is true when am is true.
∴ By Mathematical Induction an = 3.7n–1 is true for all natural numbers n.