Question

1. Exercise 18, section 5.4. Compute 90, 91, 92, 93, 94 and 95. Make a conjecture about the units digit of 9n where n is a positive integer. Use strong mathematical induction to prove your conjecture.

Answer 1

n is a positive integer and here 90, 91, 92, 93, 94 and 95 is 9⁰ = 1

9¹ = 9

9² = 81

9³ = 729

9⁴ = 6561

9⁵ = 59049

so mathematical induction is prove given below.

Explanation:

9⁰ = 1

9¹ = 9

9² = 81

9³ = 729

9⁴ = 6561

9⁵ = 59049

lets us consider by P (n) for the base case n = 0 then 9⁰ = 1 So P (1) is true

k ≥ 1so consider all integers

1≤l≤ k

There is a need to prove P (k+1)

if k is odd

`9^k^+^1` =
`9^k`.9.

`9^k` is 9 so

`9^k` = 9 +10m

then

`9^k^+^1` = 9 ( 9 + 10m)

=81 + 10 (9m)

so digit is 1 and k +1 is even