Question

For every positive integer n, prove that 7 n - 3 n is divisible by 4 [ Please explain with steps because I did not understand the NCERT Example in the textbook]

Answer 1

because...f(n+1)-f(n) is divisible by 4..........so  f(n+1)....is divisible by 4...because...f(n) is divisible......by 4.....it is a process
mathematical .....induction

Answer 2

Answer:

We can write

P(n):7n−3n is divisible by 4.

We note that

P(1):71−31=4 which is divisible by 4. Thus P(n) is true for n=1

Let P(k) be true for some natural number k

i.e., P(k):7k−3k is divisible by 4.

We can write 7k−3k=4d, where d∈N.

Now, we wish to prove that P(k+1) is true whenever P(k) is true.

Now, 7(k+1)−3(k+1)=7(k+1)−7.3k+7.3k−3(k+1)

=7(7k−3k)+(7−3)3k=7(4d)+(7−3)3k

=7(4d)+4.3k=4(7d+3k)

From the last line, we see that 7(k+1)−3(k+1) is divisible by 4. Thus P(k+1) is true when P(k) is true. Therefore, by principle of mathematical induction the statement is true for every positive integer n.