Binomial-Theorem-And-Its-Simple-Applications Question 229
Question: For every positive integer n, $ 7^{n}-3^{n} $ is divisible by
Options:
A) 7
B) 3
C) 4
D) 5
Correct Answer: CShow Answer
Answer:
Solution:
$ ={7^{(k+1)}}-{{7.3}^{k}}+{{7.3}^{k}}-{3^{(k+1)}} $
$ =7(7^{k}-3^{k})+(7-3)3^{k}=7(4d)+{{4.3}^{k}} $ [Using (i)]
$ =4(7d+3^{k}), $ Which is divisible by 4.
Thus, $ P(k+1) $ is true whenever $ P(k) $ is true. Therefore, by the principle of mathematical induction the statement is true for every positive integer n.
BETA
