Question

Consider the following sequence of successive numbers of the 2k-th power:
1, 2^2k, 3^2k, 4^2k, 5^2k, ...
Show that the difference between the numbers in this sequence is odd for all k ∈ N.

Answer 1

Given  : 2^2k, 3^2k, 4^2k, 5^2k, ...

To Find : Show that the difference between the numbers in this sequence is odd for all k ∈ N.

Solution:

2^2k, 3^2k, 4^2k, 5^2k, ...

Any even number raised to power any +ve integer  is always even

Any odd number raised to power any +ve integer  is always odd

Hence two consecutive numbers on sequence are

EVEN & ODD

Difference between two odd number = EVEN

Difference between two even number = EVEN

Difference between  ODD & EVEN number = ODD

Hence the difference between the numbers in this sequence is odd for all k ∈ N.

Learn More

using euclid's division Lemma Prove the difference of squares of two ...

https://brainly.in/question/16817346

https://brainly.in/question/28235027

Answer 2

HELLO DEAR,

GIVEN:- There are given a sequence of successive numbers

1, 2^2k , 3^2k , 4^2k , 5^2k,....

To show that the difference between the numbers in

this sequence is odd for all k ∈ N.

SOLUTION:- we, know that difference of two even number is even

example- 4 - 2 = 2

10- 6 = 4

And difference of two odd numbers is even

example - 7- 3 = 4

13 - 11 = 2

And also the difference of odd and even number is odd

example- 13 - 4 = 9

7 - 2 = 5

And positive power on odd number is odd

positive power of even number is even

So, if we find the difference of above sequence then their is also came a odd as

and a even number ,

Therefore , it gives odd number .

Hence ,proved.

I HOPE IT'S HELP YOU DEAR,

THANKS.